There were a couple of threads a while back about TX/FL/VA residency for military folks. None of them have state income taxes, so it's just a matter of your convenience. I know it's very easy in FL for military - they do it all the time.
Big thing, as mentioned, is GO TO PSD! Otherwise the Navy will keep sending your tax money to your old state.
As a heads-up, one weird thing FL does when transferring your car title/registration: if you have a car loan, when you go in and fill out the paperwork, they will send off the title request to the leinholder and you have to wait for it to come back before you can go get your new plates, etc. For well-oiled machines like USAA, it's maybe a week; small-time credit unions and such can take a lot longer.
Just saying, don't wait until the day before your old out-of-state plates expire to go get the new Florida ones.