Train connection from Rome International Airport to Florence

Yes you can buy tickets to any destination from FCO.
There are more direct trains from Rome Termini to Florence though

At the FCO train station, tell them your destination is Florence and they will sell you tickets for the next train departing for Florence. It could be the direct train or a train connecting at either Roma Termini or Roma Tiburtina. If it's not the direct train and you prefer to change at Termini rather than Tiburtina, mention that too. There are usually at least 2-3 connections per hour from FCO to Florence.

Fiumicino Aeroporto is provided with a ticket office, so you can get your tickets at your arrival. You can get a high speed train from Rome (Termini + Tiburtina) to Florence (SMN) every 10-15 minutes (including italo), and trains for either Termini or Tiburtina leave every 15-20 mins.
The best would be getting your high speed train ticket with 30-45 days in advance, so that you are more likely to pay a low price (there are high speed trains from Rome to Florence at €10 or less!).
The ticket from FCO to Tiburtina has a cost of €8. Leonardo from FCO to termini will cost €14. You can travel from Termini to Tiburtina with the subway, B line (approx 6 mins of travel + 2-9 mins of wait).
You won't necessarily find more trains from Termini rather than Tiburtina, since some high speed trains only stop in one of the two stations.
Hope this will help you ;-)

HS,

Unless you connect with one of the direct high speed Freccia trains from FCO to Firenze SM Novella, you'll be using a Regionale from the airport to either Termini or Tiburtina stations. You'll need to board the train that matches the ticket you bought, and most importantly you MUST validate (time & date stamp) your ticket prior to boarding the train on the day of travel. That applies to any Regionale train with locally purchased tickets, and also to Buses, Metro, etc.

Thanks for all the helpful advice.
My follow up question is : Since I cannot control my actual arrival time due to possible flight delay and time to clear custom. If I take advantage of cost saving and buy my high speed train ticket from Rome to Florence in advance. Am I able to exchange it for a later train just in case I miss it?

"Since I cannot control my actual arrival time due to possible flight delay and time to clear custom. If I take advantage of cost saving and buy my high speed train ticket from Rome to Florence in advance. Am I able to exchange it for a later train just in case I miss it?"

If you've bought a discount ticket, it is either totally non-exchangeable and non-refundable (Super economy), or you can apply its value to another ticket (Economy). You must be in possession of a ticket for the EXACT train you are taking BEFORE you board, or you will get a large fine.

So, most of us disagree with andrea c. Precisely because you cannot control you actual time of arrival at the train station, as you say, most of us recommend that for this particular ride, you wait to buy your ticket at the airport train station. As said above, you can buy the full ticket at this station, whether it's for one train or two separate trains.

A technical note: In Europe, customs is just a matter of walking through the "Green Channel Nothing to Declare" line. Unless you're "randomly selected for additional screening" (possible but rare), you're done - so this doesn't take any time. It's immigration (passport control) that takes time. How long it takes depends on how many other planes are arriving and how many agents are on duty, and it cannot be predicted. While North Americans often say "customs" to mean both customs and immigration, in Europe the terms are used correctly.