Hi there and welcome in Tuscany! Benvenuti in Toscana!
There are so many traditional dishes you can eat that I don't know how to write all of them.
So, may be I can suggest some places where you will taste the authentic Tuscan recipes: "Bar Ucci" in Volpaia in Chianti. Taste the Ribollita Soup, for example, wow, it's so delicious!
if you want to discover something different in Florence try the Beppa Fioraia, I think it's the best choice in florence.