The Florence train station near the center of town is Firenze Stazione Santa Maria Novella (Firenze SMN). Coming from town, here is the landmark
You go under the very busy intersection, through an underground shopping mall, and back up to the station
Notice the signs
And yes! There is actually a ticket counter! I don't know how we missed it on our earlier trip - there is even an information booth - I talked to a nice young woman (in English and a smattering of Italian) about my ticket.
Boards show the trains, scheduled times, how late they are, and, eventually, which track (platforms/binari) they are on
Make sure you are watching for Departures, not Arrivals! ;-)
You can also check this permanent board for normal train schedules and tracks (high-speed and regional)
This is our prepaid high-speed ticket to Venice (the card is our vaporetto ticket that we got in Venice). For high-speed trains, this is all you need, and you don't have to validate it before boarding - just show it to the on-board conductor if he asks for it. There are a few employees around if you have to ask questions, but the cars are clearly marked, as are the seats. You should expect to board 10 minutes before the scheduled departure time. Once on board, you can change seats (but of course, if someone has that seat, you'll have to move).
Regional trains (ie, not high-speed) are a little different. You buy the ticket on the spot, either from a booth or at the counter. Then you validate it at a machine near the track (any of them will do), board your train, and take a seat.
Both types are easy (especially after you've done it once!)