I prefer to call the city Firenze, like the Italians do, and we do in Denmark. As many other Italian cities it’s is crowded by tourists, but a guide told us that if you come in November, January or February there will be few tourists.
We did what you are supposed to do, visit the Uffizi galleries and watched the magnificent medieval art. Some of it is beautiful and has its value as art preserved even today, but a lot of it it simply to crude and primitive to my taste e.g. all the religious stuff I can be without. When I visit a museum and stumble into a hall with religious art from the middle age I will usually haste through the hall to get to something interesting.
The town hall is a landmark in the city, next to the Uffizi gallery, and I will recommend you to visit the tower. You have a magnificent view of the city from the top, and curiously even though we visited it during the day there were rather few others, so we could have an unspoiled experience of the scenery.
To see other art than the medieval stuff is also possible in Florence, as there is more than just one contemporary art museum. The Museo Novecento is a nice little example, as you can see the collection in 1-2 hours. When I visited the showed mostly Italian artists – of whom I know few – but I have seen works of Piero Manzoni in other places.
To experience other sides than the usual tourist stuff we joined a graffiti tour, booked through AirBnB. Please read more of this at my wife’s site(it’s in Danish).