My husband and I stayed at Lorenzo's apartment for 4 nights. The location is amazing and even though it is in the heart of tourist activity due to proximity to the Duomo, we found small streets with small grocers to pick up food to take back to the apartment.
One of our favorite days was taking the #7 bus from Piazza San Marco to Fiesole. If you go to the last stop you arrive in the town center. We accidentally got off at the first stop in Fiesole, but it turned out to be the perfect choice for us. A very kind woman told us to take the smaller road to the left for a beautiful walk up to Fiesole. It was a hike, but we were by ourselves and she was on target with her advice. Toward the very top there is a bench with an overlook and the road choice to the left takes you into the center. There was a Saturday market in the square and a wedding at the church. Do some research on Fiesole and ask Lorenzo. Turns out he is a runner and regularly runs to and from Fiesole! He's very warm and personable and has lived here, as has his wife, since childhood.