Part 1: If you buy a car at a dealer don't let them charge you sales tax.
You'll need from the dealer
a) Title (if you get it with the car, great. Shaves weeks off the process)
b) Temporary plates good for at least two months if not title in hand
c) VIN verification letter on dealer letterhead. Otherwise you can have base cops or any john law verify it. Make the dealer work for that exorbitant doc prep fee.
Some states make dealers charge you sales tax by law (CA, for example) unless you take delivery of the car outside the state. Others like OR, MT charge no sales tax. TX doesn't charge any for vehicles taken out of state and that's what I did. You obviously cannot buy a car in FL. Buying from private persons is really no different except they won't charge you sales tax, and you have to take care of the temp plates and VIN verification yourself.
Part 2. Download the milpak.pdf and familiarize yourself with the contents. You're gonna need the title app filled out, a copy of your Mil ID, FL DL, proof of insurance, a check for the funds and then the all important tax exemption form and maybe your orders. Anyone with a notary stamp can hook you up with the exemption form, yeah most of us are notaries and don't even know it but neither do most of the DMV workers out there. Throw in the stuff you got from the dealer and express mail it to your county tax collector. I mailed a return envelope in it since my paper plates were running out. It also helped having signature required when I had to track my packet down after a week of no action.
The biggest sticking point in all of it was waiting for the title. If there was a previous lien on the car, it'll take 4-6 weeks and those paper plates come in handy. We bought the car in late feb, got the title in late march and the plates in mid april. Plenty of opportunity for streamlining especially in the time required for the title to arrive and the process time at the DMV. My high caliber individual of a dealership customer adviser would not mail the title to anything but my FL address on my DL, but the purchase agreement and everything else went just fine to our CA duty station address. That took an extra week.
If you have a FL plate laying around it is also possible to transfer that plate onto your new car, provided no one has claimed the plate.
I did that and saved the $225 initial registration fee. If you were a FL resident when you enlisted, that fee is waived entirely even if they issue a new plate and initial registration. My grand total was $175 plus 30 bucks in express mail envelopes. It's legit, they're up front with it and an awesome deal.