You'll find wine festivals in Florence more often than in any European city, and during the summer they run almost every week. In addition, every restaurant, every cafe, sometimes every vendor on the street sells Florence's best wines.
These grapes give Chianti a unique flavor, with a fruity aroma and spicy undernotes. Most Chiantis only have 70% Sangiovese grapes, with a blend of white grapes as the other ingredients, but all are of high quality. Chianti is inexpensive as wines go, and it goes down smooth and easily.
If you want a finer Chianti, you can pick up a Chianti Riserva, which are aged for 38 months rather than the traditional four to seven months. Drier Chiantis are labeled Chianti Superiore (but the Chianti Classico is never labeled Superiore).
Trebbiano toscano is a sweet white grape, and it and malvasia grapes are the typical blend added to chiantis that aren't pure Sangiovese. Trebbianos are often converted into brandies, which is good as the wine itself is very mildly flavored and does not keep well. If you're looking for a wine that keeps well and that you can transport home, don't go for this one; instead, have it for dinner in Florence with your fish or chicken dishes.
Pomino Vin Santo is the most popular of the trebbiano-based wines in Tuscany, and it is a white wine blend of trebbiano and malvasia grapes. It's a dry wine with a fruity, nutty flavor.
Brunello di Montalcinos are released as normale or riserva; normale are aged 50 months, riserva for about 62, and the wine can improve with age for as long as 20 years, sometimes longer.
Beyond these, you'll find dozens of excellent wines in Florence. Be adventurous when you're here, and try something new with each meal.