OP, what time of year are you going? This will help us provide some suggestions beyond the basics.
There's plenty to do around Firenze besides museums and galleries (although them, plus churches, are the crux of the city), but I do recommend spending a considerable amount of time in them. The Bargello is often overlooked in the tourist rush. I'm a big fan of this old school prison. For views of the city without taking a bus to Fiesole, I recommend a walk up to Piazzale Michaelangelo. You may choose the winding roadways up or the back way via steps at the base of the hill behind apartments. Cross over the Ponte alle Grazie and head east along the river. You'll find it if you're adventurous enough. A climb up the Campanile or Duomo is a must as well. Ross King wrote a great book (IMO) entitled 'Brunelleschi's Dome'. If you have the time and are a reader at heart I recommend taking this on before visiting the Duomo.
Check out the Mercato Centrale to stock that picnic basket. A walk around there is an experience in itself. If the Viola are in town, assuming it's football season, go see a match at the Artemio Franchi. It's a 30-45 minute walk from the city center over by the 'Campo di Marte' train station.
A considerable amount of time should be spent enjoying apertivo and the countless hole-in-the-wall food & vino places.
If you need any help with Rome I'm sure you can find it here as well. Just let us know!