Applied online for several city and rural carrier positions in my area. There is an online test, and if you get through that, they schedule you for a computer based test at a testing center. Mine was at a nearby college with 4 other people. It was primarily memory and pattern recognition - no real way to study, either it's a skill you have or not. Just get enough sleep and be alert and ready to go.
Test results were in my email immediately after checking out of the test center. I scored an 86, passing I believe is 70. From there, things go in varying order, depending on the postal center you're applying to. I received emails for a background screen and drug screen shortly after the test (2-3 days, maybe). I completed these things and then began to receive requests for interviews.
Interviews are supposed to be standard; I had 2 where the interviewers stuck to the list of things they are supposed to ask, and one where the guy was completely nuts - he cussed up a storm and went on about himself, how you can do anything you want if you move up in the system, and how much he hates Amazon and iPhones. It was the most bizarre and completely unprofessional interview I've ever had. The interviewer then either "strongly recommends", "recommends", or "does not recommend" you (it's literally a set of boxes and they check one). If they recommend you, they will give you a fingerprint card to take to any police or sheriff office to be fingerprinted.
Within 10 days of the 2nd interview (the one with the crazy guy), I received an offer from them via email. They give you 7 days to either respond or decline via an online form. I had my 3rd interview the day after I received my offer from the 2nd one, so I held out on responding right away to the first offer, as the 3rd location is closest to my home.