Trains from FCO to Florence

You don't need to purchase these before your arrival; your flight might be delayed causing you to miss the train.

You could go from FCO to Termini and then to Firenze SMN station or you could use the Fiumicino Express bus that leaves from the bus station at FCO and heads to SMN bus station which is adjacent to the train station. We use this bus when we went from FCO to Napoli in September and it was great- clean, quiet and convenient. It saved us time and money and avoided 2 separate trains. We did not make reservations and glad we didn't, as our plane had a strong tailwind and we arrived 1.5 hours ahead of schedule, which was a first for us in the many times we have flown into FCO. We paid the bus driver, he packed our luggage and we were on our way enjoying the beautiful countryside as we drove down.

Buy at the Airport station. That way you know you will make the train. You never know with air travel and weather related delays.

You don’t say when you arrive but there are two high speed Frecciargento trains that originate from the airport. If you catch one of them, you don’t need to change at Roma Termini or Tiburtina.
One departs from the airport at 11:08, the other at 15:08. They both go to Venice (Venezia Santa Lucia) passing through Florence and Bologna of course.