You can definitely buy your train tickets from FCO all the way to Florence, there are a lot of trains but you should check which one is the most convenient to you. There are local regional trains that might leave Termini earlier and take longer to arrive in Florence because they make lots of stops. I would recommend catching a fast train (Frecciarossa or Frecciargento) trains from Termini to Florence, with no stops or just one. You likely will get one ticket from the airport to Termini, then it is one ticket from Termini to Florence. The fast trains require you choose the date, time and seat while slower trains might not actually give you an assigned seat, you just climb and sit down wherever you want. At all times make sure your luggage is within your sight, do not leave them at the start or end of the wagon because those are targeted by thieves, especially when the trains are full.
One train station, all works the same, doesn't matter on the train provider although in Italy there are only two: Trenitalia and Italo. If you're buying tickets online, go to these websites. Any other is a middle man and likely makes you pay more for the tickets.
It is very easy to travel by train: at the station where the tracks are, there are large boards with all trains leaving, another for all departing trains. It will give the number of the train, destination, time its supposed to depart and the track number (called "binario" in Italian so look for that word). You head to the track and walk down the pathway to be sort of at the middle of the train, if it isn't there yet, you wait. If you have assigned seats, your ticket will give you the number of the wagon and the seat number, all the wagons have the number on the outside. So it is easiest to look for the right wagon first from the track, then only get on at that point and look for your seat number.
I hope this is clear enough, if you have any questions, feel free to post them!
