Rome to Florence

Train. www.Trenitalia.com is the website with an English option to research schedules. There are trains from FCO to Florence a couple of times a day, but you can easily take the Leonardo Express to Rome's main train station, Termini, and change there to a high-speed train to Florence. (Note it is "Firenze" on the Trenitalia site.) The Leonardo Express runs about every half hour and there is at least an hourly train from Roma Termini to Firenze Santa Maria Novella, the main station there.

We usually recommend not buying a ticket for your arrival day as your flight could be late. Buy a ticket when you have cleared passport control and customs.

In addition to Laurel's good advice, check the schedules for Italy's other (but more limited) rail line, Italo.

http://www.italotreno.it/en

While they don't have any trains from Fiumicino, they do from central Rome (Termini or Tiburtina stations). Prices are often competitive. As she pointed out, you can get to Termini via Leonardo Express train, and there are inexpensive shuttle buses to both stations as well although they're slower.

http://www.trenitalia.com/tcom-en/Services/Fiumicino-Airport

http://www.adr.it/web/aeroporti-di-roma-en-/pax-fco-bus

Just go with convenience. It is nice if you can catch one of the two trains a day to directly to Florence (it is not non-stop) but changing trains in Termini is not a big deal. Buy you tickets at the train ticket office at the airport. Take the L Express in and walk to the next train. Even with luggage it is piece of cake.

The other possibility is to fly to Florence since you are at the airport especially if you can book your flight to Florence as a part of you total ticket to Italy.

We had the same problem two years ago. I invested 50 euro for a driver, from FCO to Termini (the main Rome rail station). It was a worthwhile investment, as I'm not at my best after a 9 hour red eye flight from the States & it was the wrong time to figure out foreign bus/subway journeys. The ride across Roma was thrill packed...strong confirmation that I chose correctly, to trust the ride to our young driver, who clearly was channeling Mario Andretti. He dropped us at a side entry to the big Termini Station that was only steps away from our platform for the "Rossa Freccia", the high speed train to Firenze. The train was perfect...2 hours through the Tuscan countryside in comfort, with snacks, drinks, & big windows. I wish our flight over the Atlantic were half as comfy. The private car/Rossa Freccia combo was a bit spendy for us, but settling into our flat in Firenze, I looked back at it as money well spent.

If you go to Florence immediately after landing, don't pre-purchase a ticket, because you don't know if the flight will be on time or how long the immigration/baggage retrieval procedures will take. Just buy tickets at the station across Terminal T3.
There are two trains a day that depart directly from the airport station (Fiumicino Aeroporto) and go all the way to Venezia (Venice), therefore passing through Florence (Firenze). They are at 11:08 and 15:08.

Outside of those two, you will need to change trains at ROMA TIBURTINA, where you can catch the high speed train going north to Florence.
From the FIUMICINO AEROPORTO station to ROMA TIBURTINA you'd take a regional train.

Be aware that the displays show the number of the train and the last destination. Therefore you are unlikely to see FIRENZE SMN in the displays because most trains continue past Florence (generally the last destination is likely to be MILANO or VENEZIA), in any case any high speed train going north from ROMA TIBURTINA is also stopping in Florence (FIRENZE SMN).

I do not recommend that you take the Leonardo Express to Termini to catch the train to Florence. Taking the regional trains to Tiburtina will save you money (the Leonardo Express is twice as expensive), and you will board the exact same train, because the regional train takes a short cut. Also Tiburtina is smaller, easier to navigate, more modern, and has less hasslers and pickpockets. Also be aware that from Termini that are some trains going to Milan that do not stop in Florence, they go straight to Bologna and Milan. So if you accidentally board one of those, you won't get off in Florence. At the Tiburtina station you are assured to board a train stopping in Florence. All trains going to Florence stop at both Roma Termini and Roma Tiburtina (all of them).