Most of my favorite spots in Florence are across the river in the Oltrarno neighborhood.
Trattoria Cammillo (Borgo San Jacopo)
A slightly bustling but homely old-school restaurant. Delightful service. Have the tagliata.
I Quattro Leoni (Piazza della Passera)
This has become quite famous in recent years, celebrity photos now line the walls. But the food is great: I recommend the fiocchette alla pera.
Trattoria Angiolino (Via Santo Spirito)
Rustic and affordable spot where you can watch your food being cooked.
Casalinga (Via dei Michelozzi)
Canteen-like diner which used to be full of Florentines on their lunch break. now open also at dinner. Cheap and a fun experience.
Caffe Megara (Via della Spada)
My favorite place for lunch. You can also watch Italian soccer on a Sunday afternoon.
Florence is not a great town for pizza: I boycott Il Pizzaiuolo for personal reasons. Better options include Baldovino and Yellow.
Search on this board for my post entitled "Florence Report - December 2007" for a list of good places to try in Florence along with write-ups of what we ate. I strongly suggest Teatro del Sale for a unique meal, but you will see my other recommendations (with prices we paid 12/07). Missing from that post are Sostanza for bistecca fiorentina and chicken in butter sauce, Le Mossacce between the Duomo and the Bargello for a quick sit-down lunch with the locals, and a few others good for a nice lunch or light dinner. But start with my post, then search for other listings here on Florence restaurants.