Very few tourists think about going to the other side of the Arno to check out the nightlife in Florence. It's true that there isn't a large selection of clubs available for dancing, but when you are into a more chill atmosphere and you want to hear some fantastic live music, this is the place to go. You'll find delicious, specialized cocktails and artisan brews that can be hard to find in the more tourist-centric city center.
After crossing the Arno River, make sure to visit the Piazza Santo Spirito, where you will find some of the local staples. Volume has some amazing drinks for a good price, and there's no better view of the Piazza than sitting in their outdoor seating or taking your drink out and sitting on the steps of the church.
You should also consider checking out the Liberia Cafe la Cite where you can find some amazing live music and a variety of drinks available. You'll find coffee and tea along with an amazing selection of books that you might want to consider purchasing. Local artists come here to show off their musical skills and to meet with some of the locals here--you'll find music and books lovers crowded here.