How do people become Buddhist, what is the process?
There is no “process”. But you need to adhere to the 5 precepts and “take refuge in the Buddha, Dharma and Sangha” (the “triple gem” as it is called). These are typically things you do with the assistance of a Buddhist monk or nun in a monastery or in a monastic setting but mostly this is an undertaking or a commitment you make with yourself to take a certain outlook on life. You are not called upon to account to anyone but yourself for this undertaking except if you become monastic or live in a monastery.