Linda, you can easily buy the tickets at the Rome-FCO airport. The train station is in front of Terminal 3. If you take the Leonardo Express, this is a 30-minute ride to Rome Termini. Just make sure you have at least 15 minutes to change trains as the LE comes in on tracks that are a short walk to the high-speed tracks. Plus, you have to scan the large digital sign board to find your train. Look for your departure time, train number and track number (Binario number). Don't look for Florence as the destination as the destination on the sign and train may not say Florence. Italian trains show the final destination of the run. Some will say Milano and some will say Venezia. All will stop at Florence.
As for discounted tickets, you can do this. 19Euro vs 45Euro. But, the ticket is non-refundable and we don't advise this when a plane arrival is involved as the plane could be late. No way to change your ticket. However, if you buy a ticket that departs about 3hrs after your plane arrives, you have a good chance of not missing your train.