If you are looking for a way to get to know the Italian lifestyle of today or if you just want to “fit in”, the best piece of advice I can give you is to get some “aperitivo“. Originally, the word aperitivo used to mean a light drink (e.g. white wine, bellini, spritz, etc..) which was served right before a meal, usually directly at the bar, in order to open (“aprire“) your stomach before dinner. Over time, aperitivo became cool and many bars began to serve it along with some finger food such as chips and peanuts. Nowadays, the aperitivo-culture is wide-spread and most of the bars serve aperitivo with such a wide choice of food that you can almost have dinner right there. This is good, especially because in Florence you can hardly have dinner in a restaurant/pizzeria for less than €15/€20, but this way you can have it for the price of an overpriced drink.
One of the best aperitivo you can have in Florence, in my opinion, is at Kitsch bar. I don’t really like the music they sometimes play and sometimes it can be very crowded, but the food is very good and the choice is extremely wide. I would recommend to go there between 19:00 and 20:30 if you want to be sure to find a seat. Once you sit you can order a drink, specifying you want to have aperitivo, then take a plate and hit the buffet as many times as you wish.