I'm well aware that there are many different types of Buddhism, so I'm looking for the fortune-cookie sized description of the various kinds of induction/process/rituals/ceremony involved for the lay follower.
For Theravada and some Mahayana traditions becoming a lay Buddhist is as simple as taking refuge in the Triple Gem of the Buddha, Dhamma, Sangha and taking the 5 precepts. This is usually at a ceremony and "given" by the monks however the true taking of refuge is something done in the mind, not recited or given.
You are in short taking refuge, showing trust in the Buddha and his awakening, the dhamma as the truth, and the sangha as the keepers of that path, more specifically the aryan(awakened) Sangha and also all monastics. Then you attempt to live a more skillful life that is beneficial to yourself and others by following the precepts to the best of your ability.
The ceremony usually involves a recitation in the Pali for Theravadan or the cultural language for the Mahayana.
This link to Bhikkhu Bodhi's article on the subject is great for further reading on the subject.
Going for refuge, recited x3
Buddham saranam gacchami I go for refuge to the Buddha;
Dhammam saranam gacchami I go for refuge to the Dhamma;
Sangham saranam gacchami I go for refuge to the Sangha.
(1) the training rule of abstaining from taking life;
(2) the training rule of abstaining from taking what is not given;
(3) the training rule of abstaining from sexual misconduct;
(4) the training rule of abstaining from false speech; and
(5) the training rule of abstaining from fermented and distilled intoxicants which are the basis for heedlessness.
I practice within the Triratna Buddhist community. For us someone becomes a lay follower when they become a Mitra. The requirements for this are fairly light really. Self identity as a Buddhist to some extent, be working on the five precepts and be wanting to practice in the triratna context for a bit.
The ceremony itself is always described as a simple ceremony when I attend. The new mitra presents a flower, candle and some incense to the shrine. Because we are quite a talky movement the simple ceremony normally lasts for over two and a half hours and I miss my train.
There are no necessary rituals or induction processes. You can just become a student and practitioner. Being a Buddhist means constantly going through an induction process. There's an old story from the 1950s. perhaps apocryphal, about an American who travelled across the world to meet the Dalai Lama and become a Buddhist. After overcoming all the obstacles he succeeded in gaining an audience and pleaded with the DL to make him a Buddhist. Okay, said the DL, you're a Buddhist.
Practice, study... then more practice.
as far as Theravada is concerned (which i personally recommend) Yuttadhammo has a WHOLE lot of great insight and info on his youtube account! and also at Sirimangalo.org
Theravada is very straight forward and doesnt delve too deep into any mysticism or guesswork.
There are no various kinds of becoming a follower of the Buddha, householder MatthewMartin.
By going for refuge, seeking refuge on it, toward the Sublime Buddha, the sublime Dhamma, and the Sangha of Bhikkhus, one becomes a "Buddhist", Upasaka, Upasika (householder having the Gems as his/her refuge) or more if asking for leaving home under the Tripple Gems (the second might not for all accessible, give-able).
The become a follower of the Buddha, Upasaka, Upasika, it requires to perceive the real Buddha, to perceive his good Dhamma, to perceive his straight following Sangha, and to for refuge in them, declaring it by body, signs and thought possible best toward the carrier of his heritage.
What ever else refuge, either wrong perceived or incomplete can not be seen as the entrance of Upasaka-ship.
If going outwardly for refuge toward "fake" Gems, as long as perceived as proper and no doubt, there would be no problem.
If having gone to refuge to the Tripple Gems and having doubt later, it's good to take anew refuge toward proper objects. If one has the wrong refuge taken the Sangha might be not so able to guide toward right refuge.
If having gone to refuge toward the right Tripple Gems outwardly but still perceiving them in wrong ways, the refuge as been taken correct to that extant and the Sangha would be more able to help to derive also to correct understanding.
It's worthy to note that, in cases where one has not reached the stream yet (having gained firm confidence, no more relaying on something outwardly the Tripple Gems), as soon as this refuge has gone broken by relaying on something else, it should be renewed since it is not secure, not firm, at this stage. This being the reason, Lay-Buddhaparisata as well monks are eager to renew this refuge min. weekly of on full and half moon days, requesting it anew from the Sangha of monks, after having asked for forgiveness for fault toward the Tripple Gems.
This is in as far an important aspect because without refuge into the Gems there is no protection by the Gems for one and one is open to do a lot of unskillful deeds.
Taking refuge, better asking for refuge, is always better and more secure gained if in relation to the community of the Buddhas homeless disciples.
Note that the taking on of precepts and also declare such toward the Sangha might not only be a additional possibility for the Sangha to guide one, but also make one not only to "just" follower but to a virtuous (good) follower with more chances to gain much benefit form this undertaking.
Although there are certain formal ceremonies, rituals, practiced, they are not necessary required and monks would guide one to make it in best ways after being asked to to so, which is good to declare in a devoted manner and announcing three times. One then only needs to follow the advice, speak following them, and good Sanghas pronounce the refuge formula in two ways, so that it would fit also to Vinaya measures.
General on the problem of wrong perceiving, see also: What's the safe bet, the secure practice, to gain relation toward the right refuge?.
(Note that this gift of Dhamma is not dedicated for trade, exchange, stacks or entertainment but as a means to make merits toward release from this wheel)