If you live in Montgomery County, you go to the Sheriff's Office at 115 S. Perry St. You can Mapquest that from wherever you live in Montgomery to get there. Perry is a one way street going North. The Sherriff's Office is on your right, just after the big Baptist Church (also on the right). You can probably get away with parking in the church's parking lot since you'll only be there a few minutes. Go in the Perry St entrance and the first or second door on your left will have a sign over it for Pistol Permits. Go in and let the lady behind the counter know why you're there. She will give you a form to fill out. After you've given her the completed form, she'll hand you a business card with a number on it and tell you, "Give us a call in 3 weeks." Call them 3 weeks later (2 if you're lucky) and check on the permit. When it's ready, go back to the Sherriff's Office, hand them $20 and stand against the wall for your photo. They'll hand you your pistol permit and you'll leave a happy man.
If you live in Autauga County, the drill is similar but it's a same day affair with no picture and their permit isn't as cool looking.
You will need a valid government issued ID (driver's license, etc). I would recommend getting your AL driver's license ASAP after you move. Otherwise I think you have to take some sort of proof of residence with you. You can call the Montgomery County Sheriff's Office for details (334) 832-1337.
If you are just keeping the guns in your house or transporting them (unloaded, etc) you don't need a pistol permit. If you keep a loaded pistol in your car, you do.