Florence to Rome

Search schedules here: https://www.trenitalia.com/en.html
Use Italian names (i.e., Roma, Firenze)

When searching Trenitalia, choose not only the date but also the earliest time you want to travel. Note that both cities each have several train stations - the main ones are Firenze Santa Maria Novella and Roma Termini.

Unless your flight leaves Rome late in day you will want/need to be in Rome night before departure

In addition to searching for trains on the Trenitalia website, also investigate Italo's offerings at https://www.italotreno.it/en I prefer their high-speed train service and I've found it often cheaper than Trenitalia's. It all depends on when you book, of course, and what discounts are offered.

Assuming you are flying home from Fiumicino and not from Ciampino:

• Any day there are many high speed trains from Firenze SM Novella to Roma Tiburtina and Roma Termini stations; they are run by trenitalia.com and Italotreno.it, 2 competing companies.
• Two trains go from the downtown Rome to Fiumicino airport station: The FL1 and the Leonardo Express. Both are run by Trenitalia, Italotreno does not serve Fiumicino.
• The FL1 train departs from Roma Tiburtina station, makes 9 intermediate stops and arrives at the airport station after 47 minutes. Tickets cost 8 Euro per person, one way. FL1 Tickets not purchased online must be time stamped before getting on.
• The Leonardo Express (aka L.E.) leaves from Roma Termini and arrives at the airport after 30 minutes. No intermediate stops, 17 € per person, one way. L.E. Tickets not purchased online must be time stamped before getting on.
• Trenitalia also runs two direct high speed trains from Florence to Fiumicino. Direct means that the train makes 2 intermediate stops, but you won't have to change in Rome: Frecciargento #8583 departs from Florence Campo di Marte station at 8:26 and arrives at The Airport station at 10:22. Frecciarossa #9411 goes from Florence SM Novella (11:48) to Fiumicino Airport (14:07)