what process does one have to go through to become a priest?
It depends on what type of priest.
There are “religious” priests - ie, affiliated with a Religious Order (such as Dominican or Franciscan). These priests are accountable to the Superiors of their Order, and not directly to the Bishop of the Diocese in which they serve. A Religious Order is fairly autonomous from the local Ordinary (Bishop) unless it offers direct ministry to the lay faithful (who “belong” to the Bishop).
Then there are “Diocesan” priests, who are directly accountable to the Bishop of their Diocese. They are essentially employees of the Diocese.
The requirements for acceptance to a priestly vocation is left to the Order or the Bishop, and they vary (and even more so by nationality). There are usually educational requirements - generally the candidate would already hold a Bachelor’s Degree, and would receive a Master’s Degree at a Catholic Seminary as part of his formation (possibly paid for by his Order/Diocese). And, of course, there are other requirements. For example, a candidate must have been a practicing Catholic for some number of years (a brand-new convert would not be accepted to a vocation).
Religious Orders do not generally accept a person with the understanding that he will become a priest. He will be a brother/monk (really, a layperson), and (at the discretion of the Order) may become ordained. A Diocesan priest, of course, is expected to be a priest (he serves a six-month minimum term as a Deacon).