Qualifications
Technically, you have to be a man. But if you are not into the idea of Biblical sexism and want to be a priest anyway, talk to these ladies. With the huge priest shortage right now, the church is bound to start recognizing female priests. After all, they are allowing married men to become priests now. So you can cross chastity off of your list of requirements. That law is dead.
You will need some education, because the Bible is really difficult to understand if you aren't really smart. It's going to take you eight years altogether. You can go to college and get a bachelor's degree in something semi-related (be it philosophy, theology, history, or "Finding God in the little things" studies at your liberal arts school) and then do four years in the seminary. You can also apply right out of high school and spend eight years in the seminary. At the end of all of it, you'll end up with a Master of Divinity degree. Pretty sweet.
Once you've got that, you need to be called to Holy Orders by the bishop himself. Only then can you be ordained. It sounds like it would be difficult, but if you've gone through the educational process and weren't a total Judd Nelson about it, you're a shoe-in.
After that (and, because of all the TMZ-style scandals the church has been getting itself into), you'll undergo a background check. Then you'll spend six months as a deacon, which is basically a priest-in-training (as if those eight years weren't enough) (source).