There's another very Florentine thing to do - go eat in a hole! The Florence city center has at least ten restaurants underground that were made in the 1800 and are still open today, they were cellars were wine and simple food was served (but mostly wine). I recommend two:
Ristorante Buca Lapi (1879) "A picturesque restaurant located in the Renaissance cellars of Palazzo Antinori. Here, in 1879, Orazio Lapi opened a little restaurant in the “buca” (the hole) where he served meals accompanied by wines from the Antinori estate. Over the years Buca Lapi became renowned amongst afficionados of Tuscan cuisine. In the “mythic” 1960s it was at the heart of Florence's “dolce vita” with regulars including Prince Akihito and Grace Kelly and Prince Rainier of Monaco. The walls are papered with hundreds of posters from all over the world."
Ristorante Buca Lapi Via del Trebbio 1/r
Ristorante Buca San Giovanni (1882) "Opened in 1882, Buca San Giovanni is located in medieval cellars just a couple of steps away from the Baptistery. The history of the location is steeped in legend and it has been suggested that is served as everything from the Sacristy of the Baptistery to a secret meeting place for the Rosa Croce. The interior is decorated with a 14th century fresco and period furniture. It became a temple of Florentine cuisine and guests included the Prince of Bulgary and John F. Kennedy. Since 1997 it has been under the management of the Frija brothers who offer an original menu which reinterprets traditional flavours into contemporary dishes."
Ristorante Buca S. Giovanni Piazza S. Giovanni 8