Something that i would say is try to spend a bit of time south of the river... the north is where all the obvious stuff is, but the atmosphere in the south is beautiful... so relaxed and welcoming, much less busy with tourists (from what i saw). All these lovely thin cobbled streets and delicatessens and markets etc. I would say its worth spending at least a day down there, there are a few beautiful (if slightly more understated) Brunelleschi churches to see too. We went there on our last day and i wished i had spent more time there.
Also the Giardino di Boboli (again, south) is very very much worth a visit, I had such a beautiful afternoon in there. The Pitti Palace (to which the gardens belong) is very interesting, and has a few wonderful parts to it, but it might overstretch you if you're short on time, and you can pay for entrance to the gardens seperately as I remember.
But my best memory of Florence, without the slightest shadow of doubt is our visit to the Basilica di San Miniato al Monte, a church at the top of the hills to the south of the city, you can take a slowish, beautiful (but maybe slightly tiring) walk up from around the Pitti Palace which takes about 45 minutes I think. The view from the top is absolutely stunning. We went up just before sunset and saw the entire city basked in dusty golden haze, long shadows cast across the red rooftops. The church itself is beautiful, and we were fortunate enough to be there as monks were in chant, which lilited out as we sat looking over the city. It makes me teary eyed to think of it! You can then also meander down to the Piazzale Michelangelo just down the hill, which was fun and vibrant, people busking, eating, drinking, laughing
Florence is such a wonderful place, I hope you have a great time
