Buongiorno . . .
O my goodness, you would need months to try out all the great places to eat in Florence...and another couple of weeks if you want to add on San Gimignano!!
If you just want to have fun with food try out the San Lorenzo Food Market, it will give you lots of options all under one roof...and they love to talk about their specialties.
I just had dinner at Piazza del Vino (Via della Torretta 18 | Incrocio Via Campo d'Arrigo) and I loved that the menu was divided into regions of Italy with typical dishes from each area. A bit on the pricey side...but I really enjoyed the ambiance (nice even if you are single) and the quality of the wine & food. I had a tuna steak with a pistacchio crust (a dish from the south of Italy) And it was perfect. However if you are looking for local Tuscany dishes - the ones on the menu were authentic.
The last time I went to San Gimignano, it was June and I was with my sister. We ate outside at Le Vecchie Mura. We were enthralled with the landscape views and the food was excellent. We shared a pasta dish (ravioli with cheese inside) and a tagliata (a steak with parmesan flakes on top)....and we each got our own "dolce"...I wasn't about to share that!! Reasonable price and like I said early great scenery.