absolutely yes! You can catch a train from Rome to Florence the same
morning you arrive in the airport in Rome.
>From the airport you must take a train to the central station (stazione
Termini) and than change your train to Florence. Fast trains to Florence
runs almost every hours so no problem at all.
Even though a Friday could be a busy day to travel by train, I believe
you'll find a couple of seat on the train. Moreover you'll find a seat
for sure if you travel in the 1st class.
Abou seat reservation, I wouldn't book in advance by net (and pay ?? for
the commission) and risk to lose your train if the flight is in late. You
can easely book your seat at the train ticket office AT THE AIRPORT train
station where you'll buy your ticket from airport to the city and be sure
don't lose the train and money!
Marco in Milan - Italy