La Cite

Probably one of the more unique spots to go in Florence, La Cite, is a library cafe that has a very authentic Italian vibe. With the walls lined with books to read and nooks to sit, it's an awesome environment to relax.
During the day you can expect poetry readings or art shows, while at night different local bands entertain the crowd of young, Italian hipsters and international students.
I happened to go in on a Friday around 8 pm and they had an Ethiopian-style buffet going on. This place continues to surprise me every time I go.
My suggestion is to try and arrive early if you want to get a good spot because the place gets crowded every night.
So if you're looking for something different to do in Florence that isn't crowded with tourists, check out the Libreria Cafe, La Cite!

An easy way to get there..
If you cross the Ponte Vecchio, go down two more bridges and take a left when you see La Carraria gelato (one of the best in Florence). After you finish your gelato, take your first right down that road and you will eventually see La Cite on your right.
Here is their website where you can find the schedule for the different artists that perform.
http://www.lacitelibreria.info