Our first real meal out in Florence was lunch today at the Yellow Bar (last night dinner was some bread, cheese and salami at the apartment). We had walked around for awhile and this looked like as much of a child friendly place that we came across (I noticed there was some Japanese Anime on a flat screen that the kids could watch while we ordered and waited for the food). This is probably not the most typical "Italian" restaurant you are going to find in Florence, but it was perfect for our first time with the kids in a Florentine restaurant.
My wife and I split a plate of Carbonara and a pizza Margherita, and the kids had a hamburger, a side of fries, and a bowl of pasta with butter. We had two cappuccinos after, and drank tap water. The bill was about 36€ with a 12% tip included. The best part of the meal was that everything was hot and fresh, and really delicious. We saw food go to a bunch of other tables and everything else looked good too. They have a large selection of pastas, salads, and some American style food. We will definitely be back. There are several reviews of the place on Yahoo! - reading through them it sounds like it can be a little crowded at night.