While May was full of all kinds of events (wine, cars, gelato, etc.), June is one the most special "Florentine" months of the year. The infamous Calcio Storico takes place, as well as the Festa di San Giovanni, the patron saint of Florence. Together these are two of Florence's oldest traditions.
There is more info and photos of the Calcio Storico or "Calcio in Costume" on Florence-On-Line. The official site is here.
We'll update the Festa di San Giovanni information as soon as I can find an official schedule. The day is the 24th (as always), which is a Monday this year - just not sure if that is also the night of the fireworks show (but I think it is). This is the poster from 2011 - "I fochi di San Giovanni"