You'd be shocked at how much different each ceremony is — even inside the same temple/centre — because there are a multitude of completely different ones, most of which would be completely incomprehensible for someone who came to them for the first time.
One thing is for certain: no matter how outrageously strange the 'ceremony' looks like, it's almost always one of two kinds: either a teaching or a meditation session (one could also see a teaching as a meditation session, of course). The important thing to have in mind is that 'meditation' just means 'training the mind' — mostly to be more mindful, more aware — but there are a lot of completely different types of training! Traditionally, it is said that just Siddhārtha Gautama taught 84,000 different types of training! (But there are many more)
If you'd drop out of the blue into a Tibetan Vajrayana Buddhist temple in the middle of a purification ceremony held twice per month, for example, you'd be seeing a very formal-looking temple, but exotic and garish-looking, with very strange images. Caught in the middle of it, you might be seeing a group of practitioners, old and new, some in loose clothing, some in formal red robes, some in more complicated attire, all laughing, singing (not necessarily 'sacred' chants, just popular contemporary songs), dancing, telling rude jokes, while drinking and eating meat. Dogs would be howling, cats would be sitting on people's laps, and children might be running around making a lot of noise. You'd run quickly away thinking that these people were all crazy or simply not 'serious', but just pranksters...
But in fact each and very one of those people there (except perhaps for the cats, dogs, and children) would be practicing a very complex and advanced form of meditation, using all senses — smell, taste, sound — during one of the most important purification ceremonies.
Obviously this would require a lot of explaining! And even for a very experienced practitioner of a different tradition, such a ceremony would be utterly incomprehensible (at least without a thorough understanding of what that training is supposed to be accomplishing).
Come in on the next day, and you might see the exact same people just sitting on the ground in perfect and complete silence for endless minutes, looking quite serious about it — and then come out of the temple, laughing, joking, making trivial conversation, having fun together, even with the presiding teacher. Again, that completely opposite behaviour might startle many people.
I'm obviously pointing out the extremes. Like @Tommy suggested, the best is to start with some introductory class for beginners. These will often be relatively short, perhaps half an hour to an hour, and there will be a mix of easy-to-understand explanations about easy meditation techniques (easy in the sense that they are very simple to follow — but have profound effects!), followed by short sessions of silent meditation of a few minutes, then back to a few more explanations, and so forth. As the vision of that particular school or tradition is better explained, you might feel more encouraged to attend to the more esoteric practices, and not feel they're completely insane and alien — because you've learned to understand what they are all about.
A few tips should be common to every session you attend. Shoes are off and remain outside in all cases (socks are fine). The same applies to any head gear — hats, baseball caps, scarves. It's customary to stand up when the teacher enters the room (like during Grammar School!), wait for him or her to sit down, and only then take a seat. In less formal environments the teacher will greet the students, and greeting the teacher back is naturally good etiquette. Although Buddhism seriously encourages questions about anything we don't understand (the only stupid question is the one we don't ask), it's customary to leave questions for the Q&A session, and let the teacher speak without interruption unless he or she asks for something specifically to you (like your name!). The reason for that is not merely to 'show respect'. It's actually a training! You're supposed to be learning to pay attention to what is being taught, and, when having questions, remember them until the end of the teaching session — that way, you're both training in paying attention and training your memory (that's why a 'teaching session' is also simultaneously a form of meditation — in this case, training your mind to pay attention and remembering things). Good teachers will always ask students, before they engage into so-called formal meditation practice (what is usually known in the West as 'silent meditation'), if everybody has understood the instructions correctly. At this point it's very desirable to ask questions if there are any doubts about the instructions.
Although teachers are supposed to answer anything (and most will do so gladly), it's good manners to only ask questions about what is being taught. You might read from many transcripts of teachings given by the Dalai Lama that people, during Q&A, ask His Holiness about pretty much everything, even if completely unrelated to what he has been saying. This is impolite. If you have very pressing questions about any other subject, it's better to approach the teacher for a private session. Most will gladly concede those, depending on their availability and time. It's also far less disruptive for the group. On the other hand, allowing the teacher to explain everything without interruption most often will allow them to actually answering your question further on — you'll see that happens quite a lot.
More traditional (even if informal!) meditation sessions will start with a short recitation, and end with another one. This is part of the session itself; it's not a 'waste of time' by 'going religious' or 'superstitious'. Again, you might ask the teacher — at the end, when the session is over — why they recite things at the beginning and at the end. They will gladly explain it thoroughly.
Traditionally, at least in Vajrayana Buddhism, more experienced practitioners will also offer prostrations to the teacher. It is very important that you do not imitate them — unless you know exactly why they're doing that! In Buddhism, nothing is pointless; everything is practice; everything has a very specific meaning and a purpose, and it can rationally be argued and discussed. So, if you are a beginner, you should just sit down after the teacher does, let the other practitioners do their things, and just remain silent. Sitting down in silence is not a form of disrespect; a teacher will know you're new and eager to learn; and he or she will most definitely be glad to explain you what's the whole point in those rituals.
Offerings are not necessary, but obviously they're also welcome. Again, they have a very specific meaning, and — yes, you've guessed it! — they are also a form of meditation, a form of training your mind. In this case, it's a training in generosity — letting go of your attachment to money and other goods. Anything really can be offered, but the important thing is that you're not really offering anything to gain some benefit, get something in return, or just by blindly showing some respect. The practice of generosity is mostly for your own benefit: to learn the importance of letting go of attachments. I remember that I offered my first teacher — before he formally was my teacher — a jar of pickles, because I had heard he liked spicy foods :) So what exactly is offered is irrelevant.
Usually such offerings are made in a way that nobody else is seeing. The purpose here is not to 'show off'. If you can't give anything to the teacher while everybody's watching, you can just leave your gifts anywhere in the centre or temple. Don't worry if they get stolen or not, or if the teacher doesn't get your things for some reason; the whole purpose is for you to practice generosity. Once given away freely, your deed is done. What happens afterwards with your gifts is not your concern. This is often misunderstood, because one thinks that one should have some form of 'control' to what is being done with one's gifts! I have seen teachers who take money, for instance, thank you, and almost immediately afterwards give it to someone else. That even happens with way more precious gifts — golden watches or statues or jewelry. Remember, teachers are practitioners as well, and they also practice generosity towards others!
If you are planning to ask for a private meeting with the teacher, where you will be able to ask any question, it is customary to bring a special offering just to him or her personally. This is a sign that you understand perfectly that these teachings will have a profound meaning in your life: you will learn to be more alert, pay more attention, become a better person. And because that's so precious to you, you wish to show how thankful you are for this kind teacher to be able to spend some time with you. Traditionally, in Tibet, it's customary to offer some sticks of incense, which the teacher can always use for his or her own practice (be careful not to pick Indian incense, though, because it often contains animal fats which are not appropriate as an offering — animals had to be killed to produce that incense. If you're unsure, just ask in the temple or centre for appropriate incense to be given as offerings to a teacher) and something that you know that the teacher likes to eat — biscuits are typical, perhaps some chocolate, or something like that. In the case of a private meeting, offerings are usually given at the very beginning, before you start with your questioning.
When the session ends, students stand up immediately before the teacher does the same. There are a few conventions which vary slightly. Some teachers assume that if the students don't get up, they wish to learn more, and so the teacher will continue. Others, when seeing all students suddenly stand up, might interpret that as dismissing the teacher, which might be slightly offensive. Teachers well acquainted with Westerners will often say good-bye first, so everybody knows that it's time to stand up, and they will rise slowly, to allow everybody to stand up just in time. Prostrations are never made at the end, but it's customary to remain standing, slightly bowed, hands palms touching, always turning towards the teacher, until the teacher leaves the room.
Because Buddhism has its roots in Indian culture, some things might sound silly to us Westerners, but the traditions are kept. You should never turn your back to a representation of the Buddha — or to a teacher. That shows disrespect. That doesn't mean that you need to walk backwards all the way (and risk stumbling!) if you leave a temple with some statues, but you can definitely do that for a few backward steps (safety first! tradition next!). If your teacher comes from the Far East and is not acquainted with Western traditions, it is rude to stare them directly in the eyes (by contrast, a Western teacher, or an Eastern teacher well-acquainted with our traditions, will understand that staring directly in their eyes is a sign of paying close attention to what they're saying, which would be a sign of respect in our culture).
Pointing your feet to the teacher and/or any Buddha statues is also considered very rude. For the Indian tradition, the feet, because they are touching the mud and filth on the ground, are considered 'dirty'. So pointing them at objects worthy of respect is very impolite. If you really need to stretch your legs, do it sideways, so that your feet are not pointing to anyone. In the winter it's sometimes easier, temples might distribute some blankets, and so you can cover your feet that way.
If you're a beginner, and you cannot stay cross-legged for a long period, it's fine to ask for a chair if one is available. Sitting cross-legged is not even an 'Eastern' thing. Chinese, for instance, use chairs. The Japanese 'sit' by kneeling down. So there is no fixed rule for what 'has' to be done when attending ceremonies or teachings. However, Vajrayana Buddhism has some very advanced meditation techniques which require sitting down cross-legged, which will be explained to you why — in due time. In the mean time, in the West, many temples might have some chairs, and it's neither rude, nor impolite, to ask to be allowed to sit there. In fact, if you can sit down and remain still in your chair, it's far better than be constantly stretching legs, arms, back, etc. because you cannot (yet) remain still cross-legged. It's also far less disturbing for others. When sitting down, you should not cross your legs, nor play with them, sit sideways, etc. The appropriate way is to sit down with both legs parallel to each other, feet well planted in the ground, your back straight.
Inside the centre or temple, it is very rude to speak of anything futile or trivial (except on those very exceptional occasions mentioned at the beginning!), but Buddhists are supposed to be happy, easy-going people, so they just have their fun outside. It's not 'hypocrisy' to 'behave well' inside and be a jokester outside — it's only important to understand that, one one hand, Buddhists take their practice very seriously, but the result of that practice is a way of happiness — so being too serious, too worried, too stressed, too stiff is definitely not a good sign! Buddhists are easy-going and certainly behave that way outside the centres or temples! Just look at the Dalai Lama, he's always smiling, laughing, telling jokes, amusing himself with silly things, and pretty much spreading that easy-going nature of his to everybody around him. But inside a temple he's serious.
Good teachers, however, will also teach in an informal way. This is specially the case in the Nyingma and Kagyü schools of Tibetan Buddhism. It's not unusual for a teacher to tell jokes, illustrate a complex point of doctrine with a hilarious anecdote, and, in general, make the students relaxed and in a good mood. Of course, visiting teachers who come for the first time to a specific temple will be much more formal, because they might not know what the reaction of the audience will be. You can certainly see that a teacher who has come to a centre often — or who is a resident at that centre — will most certainly have a much more informal way of teaching.
In a Vajrayana temple, all objects and images you see there are 'special', in the sense that you have to be given special teachings to understand their purpose. You're still welcome to ask about anything, but please refrain from touching anything. In particular, you should not touch anything belonging to the teacher — or to other practitioners. Vajrayana temples have musical instruments around, but, again, they're not toys, and their purpose is not really just making musical sounds — there are far more profounder meanings in the music — and they are not to be handled unless you have been given special instructions and explanations about them. This sounds a bit crazy and secretive, but you can imagine how you would feel if a visitor to your place would start opening the drawers and look at your underclothing and ask strange questions. Of course, if that visitor happens to be your sweetheart or someone very close to you, you would share a certain degree of intimacy, and then you'll let them do that to your home easily. Vajrayana Buddhism is a bit like that: once you have some 'intimacy' with the whole philosophy and the whole point of the practice, then, yes, slowly, you'll be able to touch everything, do everything... and, in fact, it will even be a requirement. In many Vajrayana temples, new visitors might be shocked that suddenly one of the other practitioners start to do the practice without a supervising teacher, or even teach some meditation techniques, etc. There is no real 'dividing line' between 'student' and 'teacher'. A teacher becomes a teacher when their own teacher tells them to do so! So, every time you see someone who is dressed just like you, very casually, all of the sudden leading a meditation session or more complex ceremony on their own, you can rest assured that they have been given explicit permission from their own teachers to do so. In fact, they're practicing to become teachers one day. And the first thing that they all learn is how to do everything — because teachers will die some time, and who will continue their work?
Books with recitations or anything related to the teachings of the Buddha should never be touched or handled without permission, and never ever placed in the ground. You should also never place anything on top of them — not even your glasses (if you use glasses to read). The reason for that is that these books contain the precious instructions that will teach you how to train your mind, to become more functional, a gentler being, a better person. Even if they're just photocopies of something scribbled down in a hurry, they're the most precious things you can find. As such, they are to be handled respectfully. If, before a meditation session, they handle out some plain printouts of the recitations, or some instructions, don't place them on the ground. They might look very simple and plain, but they're nevertheless very precious. And, for practical purposes, it also means that you are not spoiling them, making them dirty and unreadable, thus contributing that the next person who gets them to read can benefit from them as well. Not all Buddhist groups are wealthy enough to be constantly printing out things!
Most groups will also not give you their books or photocopies of the recitations, etc. This is not due to a lack of generosity of their part! One difference from Buddhism to other religions and philosophies is that you cannot learn anything from a book — just from a teacher. Books are just memory aids — things to remind you what you're supposed to do. But without a teacher to explain you what is meant, they're worthless — to anyone who doesn't understand them. As such, they're not easily given away. Instead, you should always ask the teacher for permission to get a copy. A teacher will know if you really intend to practice those instructions seriously, and, if that's the case, he will be most glad to give a copy or allow you to buy them. But if he's not sure yet that you intend to follow the practice as he instructed, then he will politely decline to allow you a copy.
Many teachers practice generosity towards their students. Some teachers are really not wealthy and give away everything; so the slightest gift is very important, and also a form of test. For example, it's rude not to accept something to eat or drink from a teacher. When he offers you food or drink, he or she is symbolically offering you the teachings of the Buddha — this is the most generous offer he or she can make to you. So even if you're full from your last meal, or even ill with some stomach disease, accept at least a tiny morsel of what the teacher is giving to you. This will give the teacher confidence in you: if you're willing to accept food from him or her, you will very likely be willing to accept teachings from them as well.
Last, but most definitely not least, but rather the contrary: Buddhism is a very thorough philosophy/way of life/religion/science of the mind with a vast amount of literature. Over 2,600 years of discussion and debate have been recorded by millions of teachers. As a result, it's profoundly logical and rational at every level. While externally there might be some rituals and some rules to observe, all of them have some logic behind you. So, the beginner's mind should be a questioning mind. Question everything. Be skeptical, but open-minded enough to ask everything. Think the answers over: do they make sense to you? Are the arguments rational and well-presented? If so, accept the answers, and turn them as if they were your own. If they don't make any sense to you, reject them. Siddhārtha Gautama himself said, quite clearly, that nobody should take his words for granted, just because he is called the Buddha. Instead, you should think them over very carefully and see if they make sense. If not, and you're still curious about Buddhism, ask away — ask as much as you can to see if you can clear away each and any doubt you have. Be wary of 'teachers' who refuse to answer your questions, or come up with some technical mumbo-jumbo which might sound impressive but really doesn't make any sense. Buddhist teachers do not try to 'impress' students, or 'convert' them. Instead, they describe methods and techniques, and expect students to ask them 'why does this work?' — but also that they are willing to experiment those techniques and see if they work for themselves. It's a very experimental philosophy: it makes little sense to read about Buddhism — except if you wish to put it into practice, which is the whole point.
Oh, I forgot to say: make sure you get a qualified teacher to ask your questions. A qualified teacher is someone who has been asked by their own teachers to teach. And you're certainly allowed to ask any teacher, 'who told you that you're allowed to teach?' Any serious, qualified teacher will have absolutely no problem in answering that question truthfully — and you're quite welcome to check if it's true!
I, for sure, am not a qualified teacher :) But I try to help when asked to the limit of my abilities; all I know was what my poor memory retained from the words of my excellent teachers. So all mistakes I make are only mine — not a flaw of the teachings of the Buddha, and most certainly no flaws from my own teachers.