Florida is very nice as far as gun rights go. You will need to become a Florida resident to get a CCW, but that in itself is relatively painless (just get a drivers license and register to vote, no state income tax either). If you're at NAS Pensacola, just drive out the front gate, and a few miles up is a Tax Collectors office, you can do it there.
Florida lets you buy long guns with no hassle whatsoever, and handguns have a 3 day waiting period. The waiting period goes away once you have a CCW. Ask any local gun shop, they'll give you the paperwork for a CCW. Took mine about 2 weeks from dropping it in the mail to getting it in my mailbox. Florida CCW's are also valid in 29 other states, which makes traveling nice and convenient.
Edit: As far as military goes, your military ID or a copy of paperwork granting you a pistol marksman ribbon counts as "proof of training". At least in Florida.