Trattoria Za-Za in Florence.
Loved this place...ate there twice. It was very close to our hotel. The first time our waitress was awesome. We ordered a 4 cheese pizza as a first dish to share, then we each ordered another plate for the main meal. The waitress asked if we like olives and said we whould add it to the pizza. I was expecting sliced olives, but the pizza came out with whole olives with the pits still in. I can't explain why, but it made a huge difference.
I wish I could remember the places we ate at in Rome and Florence. I got my ipad the week before the trip so I could have a lightweight book/computer type machine. I used and highly recommend Tripadvisor. I could use the wifi at the hotel and pull up all the restaurants in the area. Read the reviews, check the menu and easily find good restaurants. Never had a bad meal using this technique and it was easy to filter out the tourist traps and find some real gems.
I don't know how long you'll be in Florence, but I have one strong recommendation. Go to the San Lorenzo market and find the Central Market...a big building in the middle of it all. That's where most of the locals shop for food. They have butchers, bakers, produce stands. All incredible. Buy some fresh bread from a bakery. Get some great cheese and proscuitto (or whatever you like) from somewhere else. Add some fresh strawberries or grapes. Make a nice, simple sandwich and go eat on a square. Cheap, easy and incredible.
Always drink the house wine. The owners take special pride in their house wine and it is usually very good and much cheaper than other options.
I would also recommend taking a wine tour while in Florence. We did and it was an awesome experience. Tripadvisor is a great resource to find what you would like. We used Angela. She picked us up in Florence and we made a few other stops...Michelangelo's piazza, lunch in a Florence villa, a small town.