According to the suttas, samsara has no beginning. There are many references to this.
The answer to your question depends on how one understands the word "start." Etymologically, the English word means to jump, leap up, or move or spring suddenly, thus, to arise, to come into being. In accordance with the law of karma, the whole process of samsara itself must be self-starting.
This can be proved logically based on the law of karma, which requires that every cause have an effect and every effect a cause. Since samsara is time itself, the essence of time is the identity of cause and effect, thus, the moment, which is then extended in space as past and future to create the infinitely differentiated world that we experience.
However, as the illusory negation of reality samsara must inhere in the non-illusory, which is reality itself. Therefore, samsara also inheres in reality, and in that sense has a logical - but not a temporal - "origin." This "start" is the principle of karma itself, which is the inherent kinetic principle in transdual reality.
This is necessary to explain our experience of samsara. Otherwise, there would just be nothing at all. This point of identity of samsara and reality is, once again, the moment. Therefore, the answer to your question is that the start of samsara is the inherently timeless and therefore eternal moment, which paradoxically is the origin of time, space, and causality that endlessly differentiates in infinitely variegated patterns. This view of the cosmos remarkably presages certain theories in modern physics, especially the holographic theory of the universe.
Beginning of Samsara is stated by Buddha as not knowable for ordinary beings. Knowing rightly would be to not imagine in regard of its beginning.
Usually Samsara is imagined as round of re-births in the six realms, or much more detailed view as bhavacakra.
Your question has two parts so i will answer in the following order...
How to understand "Samsara" and how it works.
Sansara or Samsara is the word Lord Buddha picked to name a certain circle of death and life.
What is samsara
As we Buddhists believe as long as a being lust for existence he will be born according to his Karma.To this process we call "Sansara or Samsara".
So where is the start of samsara?
Even Lord Buddha was intrigued by the question and started to find a solution.The only way to do it is to look back into your own past to find your first life.So Lord Buddha went back countless amount of lives and understood that the "Beginning" is very ancient the time we have in a life is not enough to look back and see that life.So lord Buddha said this to the Sangha "There is no point in looking for a start,it is too old and trying to find an answer is a waste of time".And advised that no one should try to find an answer to it (Because they will not be able to find it)
So my friend that question can't be answered by any being in the universe,Because if there is something out of Lord Buddha's reach it is out of anyone's reach.
So how does samsara work?
How it works is complicated but i will give you a simplified version of it.
Samsara needs fuel like your car,As long as you give fuel to it it will keep working and you will have life after life to live.
Let me give an example from lord Buddha....
"Karma is the paddy field,Lusting is the water,vinyana is the seed"
In the real world without water a seed will not grow,likewise without "Greed or lust" you will not be born,without it there is no samsara but as long as you have it you will have samsara and you will have to suffer in whatever karma brings you.
Buddhism basically deals with the present. Dwelling the mind on past or future unnecessarily creates mental vexation. Hence, concepts like God, origin, etc. are not dealt in Buddhism. It doesn't matter why or how the samsara came into existence, what matters is that samsara does exist. Now when samsara exists and we exist, our effort should be eradicating problems that give rise to suffering. For example, because you are thinking about the origin, you are to some amount troubled by the question. So as per Buddhism you should not focus your mind on such irrelevant questions as they unnecessarily create suffering.
However, if you are really looking for answers then you should consider studying the philosophy of other religions. If you do, then you will find the answer to be God. Then the question of starting point of God will arise in your mind. If you look into science, it says Big Bang is the starting point. But in my view, Nothing is the starting point. Only that which has come to exist can have a starting point. How can that which doesn't even exist have a starting point? I know this sounds ambiguous, but it's hard to explain here in few words. You can check the following link if you like, but my findings there is based on Vedic scriptures: http://shunyavada.com/
From objective perspective, Samsara is a beginningless round of deaths and rebirths the beings are stuck in. "Beginningless" means, it has no beginning. The question you may ask: if Samsara has no beginning, and I am in it, how did I come to exist? This seems like a real puzzle.
Another piece of the puzzle, is Buddha's suggestion that despite being beginningless, Samsara has an end. The end of Samsara is attainment of Nirvana.
The usual superficial understanding of these, is to assume that one can escape Samsara, stop the round of deaths-and-rebirth, and effectively spend the rest of eternity in a state of perpetual lack of suffering.
Such answer though would omit a very important construct of Buddhism, one of "Tree Marks of Existence"
According to Buddha, the existence inevitably has characteristics of
- Selflessness (nonsubstantiality)
Which means, a state of perpetual lack of suffering would contradict the fundamental law of the universe (as presented in the Tree Marks). How could it be perpetual given the impermanence? How can suffering cease given that suffering is an intrinsic characteristic of existence? How can I spend the rest of eternity in this state if "I" (self) does not exist?
This seems like a serious contradiction. And yet Buddha insists, there is a way to solve this conundrum, there is a way to attain Nirvana.
Without giving too much away, I would like to hint at the direction the answer lies in:
Samsara begins when subject and object separate from each other. Samsara ends when subject and object are no longer separated. The answer lies in the realm of phenomenology of experience, including experience of suffering, life, and death. Good luck with your quest!