Taking the train from Florence to Fiumicino Airport/Rome

There are frequent trains, trip is about 2.5 hours, with one change - don't appear to be any direct.
Tickets not normally released until three months before - is often cheaper to buy in advance from Trenitalia.

Go to Trenitalia.com and enter the two end points in Italian:

Firenze S M Novella
Fiumicino Aeroporto

Pick a date that’s two weeks away, and you can view all of the time options. You will notice that some of the times are very long - 5+ hours. Select one of the options that’s in the 2.5 hour range.

Caveat - many of us don’t feel comfortable taking a train a distance on a day of a flight, me included. If a train is late, and you miss your flights, that’s a lot of money! Consider staying in Rome the night before your flight. If you still plan to come from Florence, give yourself extra buffer time, taking an earlier train because some of the connections at Roma Termini are 15 minutes. If that first train is delayed, there are several that go to the airport - just allow time for any confusion if you haven’t traveled much by train. You want the no-connections train option from Roma Termini to the airport.

Just FYI: Your most likely route will be "Firenze SMN" (Santa Maria Novella) the main station in Florence to Roma Termini, the main station in Rome, then taking the Leonardo Express from Termini to the airport. You can purchase as one ticket or separately depending on your plans.

Getting tickets in advance for the fast train from Florence to Rome will be cheaper, if cost is not an issue and your plans are unsettled, then tickets are available right up until departure unless the train is sold out.

What time is your flight? The further you are away, the more risk, the better to leave some wiggle room or buffer. Even some variance in the connection in Rome will add 15 minutes to your journey, and you never know how long security will take.

If your flight is in the morning, it is risky to stay in Florence the night before your flight. Lately the line had some technical problems (some due to apparently to possible sabotage), so there have been frequent delays.