I know I can take a train from Florence to Rome and then take local transport to the Rome airport, but is it possible to book a trip direct from Florence to the Rome airport? What is that Rome airport train station named?
Rome has 2 airports: Fiumicino and Ciampino. I'll assume you're wanting to travel to Fiumicino (FCO) but correct me if I'm wrong.
Referencing the Trenitalia website, there are no direct (no changes) trains from Florence's train stations to that airport: there is a change either at Roma Termini or Roma Tiburtina.
If you are looking to travel from Florence to FCO to take a flight that same day, most here would advise against such a plan.
( the assumption being that you have a North America destination; flying to somewhere in Europe might be feasible)
What is that Rome airport train station named?
Fiumicino Aeroporto
On the Trenitalia app, or at the station, yes, you can book a ticket from Florence, Firenze SMN to Fiumicino Aeroporto.
The journey will likely give you options for a fast train to either Roma Termini or Roma Tiburtina, then depending on which of those stations, the Leonardo Express from Termini, or the FL1 regional train from Tiburtina, to the airport. Overall travel time likely will determine the best option.
The OP asked:
....is it possible to book a trip direct from Florence to the Rome airport?
I'd assumed "direct" meant with no changes between the two. Otherwise, you're right that the trip can be booked on a single ticket Florence> change at Rome Termini/Tributina> Fiumicino Aeroporo.
Yes, we will be flying back to Florida from the Fiumicino international airport near Rome. We will just plan to stay in Rome overnight on the way home. Thank you all.