Okay, so DO whatever you want. You can totally ignore this post and do what you feel like. In fact I encourage that.
But DON'T come crying to me when you waited so long to try Gusta Pizza in Florence and now you can only have once.
DO: Go to Florence (Firenze) get off the bus, put you bags in your Airbnb or hotel or however you sleep at night, look up Gusta Pizza on your smartphone, and go directly there for lunch. I could tell you about all the deliciousness contained in those personal size pizzas, but the English langauge isn't romantic enough to describe Gusta Pizza. Also in Florence, go to see David in the Academia, and go to Boboli Gardens (close to Gusta Pizza so it's great to walk around after you eat.) Also here's a pro tip for you, if you like to save money, the first Sunday of each month, both Boboli gardens (and the rest of the museums) as well as David, are free. So take the money you save, and go buy more Gusta Pizza. There's also the classic Ponte Vecchio bridge (get your picture, be a tourist, it's okay.)
Go to Venice, buy a bottle of prosecco (or any wine really,) ask for plastic cups, and get your butt on a Gondola and live life large. Easily one of the best things I did during my time here. You also have to make a decision on if you want to go during Carneval if you decide to study in the spring. Carneval was really cool to see, everyone dressed up and such, but it also meant lots of people, and so we didn't get to tour St. Marks Cathedral and walking around took longer. It's up to you. We went during Carneval and I loved it, but when I come back to Venice, I definitely won't go during Carneval.
There are tons of things I would recommend, but I still have my don'ts to do as well. The only other thing I'll say is stay in Siena, at least for a couple weekends. You'll want to anyway to have a break from traveling (trust me, you will. I didn't think I would either.)
DON'TS: So take these with a grain of salt. Almost everything I did or had in Italy was excellent.
Don't take a weekend trip to Pisa. There's just not enough to do there once you see the tower. The tower is really cool, but it'll take up maybe an hour of your time, then you're stuck in a town that just doesn't seem to have much life. Go to Lucca instead, and stop in Pisa to see the tower, and continue your journey.
Don't travel too much around Europe. I know it's enticing to go see all these cool places, but it can get really expensive, and there are really cool places in Italy that are much closer, and much cheaper to get to. Plus, you came to Siena, Italy. It's a little strange to go study in a country and spend so much of your time outside that country.
Lastly, don't decide to not go to Gusta Pizza.
Go to Gusta Pizza.