Very easily possible. Go to the trenitalia site when the train times up up (they'll switch in the early fall from the summer times now available) and look up the times from Roma Termini to Florence. Then look up the Leonardo Express times from the Rome airport to Roma Termini. Armed with these, all you do is go to the ticket window for the train in the Rome airport (generally not that busy) and show the agent what trains you want. They speak english, but even so you will have the paper with you to help feel more comfortable. And its smart to throw in a few polite expressions (in italian) such as "please". Pay with your US credit card.
So if you're at the window at 2:30 for example, you'll want the 2:45 (or whatever) Leonardo train to Termini, and then the next express you can catch to Florence. Its even easier to do than it sounds here. Just don't forget to validate your ticket in the yellow box before boarding the train to avoid a possible fine.