I will give it a try at describing the process. The subject is quite vast and and scholars probably could take lifetimes looking into the history. Siddhartha Gautama, was the Buddha. He was from the Śākya tribe of India- a Prince who was troubled by wars from invaders encroaching upon his territory. He sought answers for all of the suffering he observed in the kingdom, and resolved to understand why it happened in life, and what to do about it.
A wikipedia note relates that the Śākya tribe was an oligarchic republic-like state of India. The Buddha was to the left of Ronald Reagan i.m.o.
So he left his home at age twenty-nine and sought wisdom until age thirty-five when it arrived after sitting beneath a Bodhi tree 45 days. He was practicing dhyana meditation that he thought led to spiritual truth. The result of his enlightenment was the sermon near Benares at Samath where he taught the noble eightfold path.
The Buddha was a philosopher-prince who found practical, moderate responses to existential challenges rather than purely religious, metaphysical answers to actual suffering. He could be regarded as an idealist-atheist with what some regard as a modernist perspective of life being phenomenal and so unique that it is nearly virtual reality that has no significance and vanishes into nothingness. Siddhartha regarded suffering as meaningless as life. To initiate causes was thought to bring transcendent effects. Reincarnation and cycles of rebirth and protracted suffering were problematic metaphysics that should be negated. The goal of Buddhism is nothingness; nihilist nothingness that might be accomplished with perfection of inaction, or possibly perfection of action with insight.
One might disagree that the Buddha was an atheist. Interestingly his ideas probably influenced western philosophers like Arthur Schopenhauer and even Friedrich Nietzsche (if his ideas about eternal recurrence originated somewhere besides advanced syphilitic mental disturbance). Schopenhauer thought life was created by a demiurge that made people suffer. One might infer that if life is a cycle of recurrent rebirth and suffering that some diety was sadistic or a hater of humanity. Thus it might be a good idea to escape that. So in a sense Buddhism may be viewed as travelling west as well as east and influencing more than three schools of followers with somewhat different traditions about Siddhartha’s ideas.
The teachings of Buddha as kept in the Pali (Theravada) Canon were said to have been written about the year zero however the earliest written records start in the 5th century a.d. That is enough time for the possibility of inaccuracy to be introduced in regard to what the Buddha actually said or wrote.Theravada established itself in India, Nepal, Sri Lanka, Bangladesh, Myanmar, Thailand, Cambodia, Laos, China and Vietnam..
Over the centuries Buddhism journeyed across Asia through China to Japan. The Mahayana and Theravada were two major forms that developed during the journey over centuries. The former was known as the greater basket and the latter, Thervada aka Hinayana, the lesser basket. Tibetan Buddhism retained the teachings of 8th century Vajrayana precepts.
Chan Buddhism developed in 6th century China and so did Pure Land Buddhism. The Pure Land doctrine is of a better metaphysical plane beyond the realm of the forever, intrinsically corrupt world-Universe. Zen is another school of Mahayana Buddhism. Over more centuries Buddhism journeyed to North America and California where it is popular in wine-drinking country.
Buddhism is the primary religion of the orient and has more than a half billion followers.