Florence is a city that never sleeps. You'll be able to find a place open well into the morning, and the bars are no exception. There are also plenty of different types that will keep you returning time after time--you're sure to find something for all tastes.
Consider looking into going to Mayday when you are wanting to meet some of the locals and want to talk about which places in Florence would be the best to visit. Not only are there some great drinks available for some reasonable prices, but the thoughtful interior is also part of what makes Mayday a spot to frequent.
Another spot worth walking over the Arno River for is Popcafe, where you can get a classic Florentine drink and enjoy the Piazza Santo Spirito right outside. You'll find a vibrant nightlife in this area, and you'll also have a chance to see what Florence is like when it's crowded with tourists. Also, if you are looking for the best and most plentiful aperitivo in the Santa Croce region, Moyo can't be beat. You'll leave not only satisfied with your drink, but also stuffed full of food that they have put out from their kitchen.