Quite a debate. Trying to put myself in your shoes, 1st time in Italy. That was us 5 years ago, and in 11 days, 10 nights we did 4 nights in Taormina, Sicily, 4 nights in Florence, 2 nights in Rome. Fantastic. Earlier this month, we did our last 10 days of a 17-day European tour with 5 nights in Florence at an apartment near Florence SMN train station, then 5 nights in Salerno.
If it were I, with those flights, I'd do 4 nights in Florence to begin, leaving room for day trips, although we did none of them this last trip, opting to just wander Florence. Yeah, Siena would probably be #1 choice for a day trip, but first take Thurs 5/7 and Fri 5/8 to see the Uffizi, L'Accademia & Il Duomo (if you're into that) before the tourist hordes arrive on the weekend. Like I said, we were there earlier this month and the difference between the number of weekday and weekend tourists was startling.
Then...for something different...I'd hop on a noon Fresciarossa (fast) train at SMN and get down to Salerno (with no train change at Rome or Naples), arriving before 4:00. Explore this neat port city for 3 nights, using it as a base & taking daytrips to the Amalfi Coast by bus (1 hour to Amalfi or Ravello, 2 to Positano, less overall by ferry) or less than an hour by train to Naples or Pompei, the latter of which is breathtaking & should not be missed. Food is tremendous in Campania, more fish-based than up north in Tuscany.
Finally, by rail (probably about an hour) to Rome Termini, like stated above staying near the station. 3 nights won't give you everything but again take advantage of the Wed/Thurs/Fri weekdays to see the popular sights you want. There still will be a lot of tourists but not as many as Saturday or Sunday. Finally, taxi to Rome Fiumicino (a real cluster-you-know-what of an airport, give yourself plenty of time) and home.
Enjoy! Remember, you cannot go wrong whatever path you choose...