Orthodox process:
Traditionally, there were no seminaries, and one learned theology and praxis face-to-face with another priest, or at a monastery. Nowadays, one usually goes to a seminary, spends several years there, and at the end of it comes out with a D.Div. degree. Admission to the seminary is dependent on one’s past academic performance, recommendation from one’s spiritual father, approval of the local hierarch, past experience serving the church, etc. Those wishing to be deacons have the option of attending seminary, or taking a correspondence course (at least in the US). There are five Orthodox seminaries in the US: Holy Trinity, in Jordanville, NY; St. Vladimir, in Crestwood, NY; St. Tikhon, in South Canaan, PA; Holy Cross, in Brookline, MA; and St. Herman, in Kodiak, AK.
At the seminary, one learns such things as ancient and modern liturgical languages, patristics, theology, the principles behind church music and art, and (hopefully) how to be a decent pastor. Seminaries are often attached to monasteries.
After this is done, one is examined by the hierarchy – taking into account the recommendation of one’s spiritual father, etc. The decision is up to the bishop. There are all sorts of disqualifying factors, from having shed blood through violence, to being too young, to not being whole in body (with exceptions), to having kept a concubine. Before the ordination, one will sit down with a priest and exhaustively review one’s life up to that point, to see if there’s anything in there that would disqualify one from becoming clergy. That priest will then give the bishop the thumbs up or not. If one wishes to marry and has not done so, this is the time to do it, as it’s not allowed once one reaches the level of the subdiaconate.
Once all that is done, a date for the ordination is set, and it takes place in the context of the Divine Liturgy.
Bishops are often, but not always, chosen from the ranks of the monks. All bishops must be celibate. Candidates for the episcopacy are chosen by the synod of a local (i.e. national) church.