Go to a grocery store (CO-OP is common there, you do not have to be a member) and you can stock up on any and all proraso for dirt cheap.
I think I got the 100ml tube of proraso green for ~2 euros, and pre-post for 3 euros.
Shaving stuff is easier to find in grocery stores than pharmacies there. I stumbled into one high end shaving store (which is where I found my DE blades for my stay there) but it was in Genova, not really that close to where you'll be.
Mentholy proraso cool is the perfect antidote for sweltering Italian heat.
There is so much to see and do that I would put the local aqua e sapone on the back burner. Santa Maria Novella is an obvious choice, and the many other fine Italian shaving products as well...but spend your time eating, drinking, learning and easing into one of the most wonderful places to be in the world.
Florence is my favorite city, but I love Siena even more. I have been to Tuscany a fair few times and most recently spent last July in a villa just outside of Siena with my entire family.
If you want some of what I would recommend, not shaving related, I would be glad to discuss it via PM.
I spent a week in Rome and it was fantastic! I ate everything that I could get my hands on and never had a bad meal or bad bottle of wine. I think the food was some of the best I've had abroad. Also the Italians serve fantastic coffee as a rule. Check out the vatican if you have time, Raphael's "The Transfiguration" was breathtaking.