We ate at a place called Trattoria Anita which was this little family run place in the back streets behind the Uffizi. It was full of Italian families eating there and was very good. That said food in general is good even on the main streets, we ate several times around Piazza del Duomo and it was always good.
Make sure you go see Basilica di San Lorenzo, inside is incredible. Uffizi was good but it can drag on, the Medici were quite the hoarders.
You also have to see David of Michelangelo at the Galleria dell'Accademia. Again as the Uffizi book online as it gets insanely busy but you're there, you have to see it.