In addition to what johnwilliamson062 correctly said...
It's illegal for anyone but an 01 "Dealer" FFL to send a handgun via the U.S. mail. Period.
It's a violation of postal regulations. You may find some people who claim it's legal because it doesn't show up in the federal firearms regulations, but they're not looking in the right place.
You can send a handgun out-of-state via UPS or FedEx, but their regulations require that you send it next-day air rather than ground. Generally the cheapest option is UPS "Next Day Air Saver", which doesn't guarantee AM delivery, but will save you $8-$10. However, the package will still typically cost $35-$50. Furthermore, you have to ship it from an actual company-operated hub, not
from a drop box or one of their strip-center "UPS Store" locations, which are franchises and aren't authorized to handle firearms. Hubs can generally be found at your nearest commercial airport with scheduled airline service. The obvious problem here, however, is that the airport may be 100+ miles away if you live out in the boonies.
Due to the expense, it's often cheaper to send it FFL-to-FFL because they can send it USPS Priority Mail. The difference in cost between this and Next Day Air may offset the FFL transfer fee.
Some unethical folks will tell you "well, you can lie about what's in the package and send it UPS Ground...", but this in itself is a violation of federal law.
Furthermore, although federal law allows
FFLs to accept handguns from non-FFLs, it doesn't require
them to do it. Some FFLs choose not to accept handguns from non-FFLs because they don't want to be stuck holding the bag if the gun turns out to be stolen or illegally modified.
do an FFL transfer to an out-of-state buyer. Period. Federal law is crystal-clear in this regard.
Whether you're required to do a FFL transfer to a Florida buyer is a matter of state law, and I'm not familiar with FL state law. Be aware, however, that almost every state has legal provisions requiring a seller to take reasonable steps to verify whether the buyer is eligible to own the firearm, and many have eligibility requirements that go somewhat beyond federal law. (For instance, TX prohibits selling to someone who is visibly intoxicated.) It's so easy to fake a faxed, mailed, or scanned copy of the buyer's ID that there's simply no way that I would mail a firearm to an in-state buyer who isn't personally known to me.
I am not an attorney, nor do I play one on TV. This is not official legal advice. Caveat emptor.