I hope you go as early as possible.So many tourists around Easter and afterwards.
Plenty of battles in Tuscany during medieval and renaissance times: every city was its own state, and your neighbour was your worst enemy. There's still a saying in Florence and Lucca: "better someone dead at home than a Pisan at your doorstep". Luckily nowadays this rivalry is mostly seen at football (soccer) matches only.
If you have a car check out Poppi in the Casentino region. There is a castle there with a plastic of the nearby 1289 battle of Campaldino, in which young Dante took part. link Plenty of medieval castles there.
Mirliton the producer of miniatures is in Tavarnuzze, a couple of kilometers South of Florence. mirliton.it
In Calenzano, to the Northwest, there's a Historical Miniature Museum link
Hopefully there will be an event during your stay that will let you see the Florentine historical procession, which gives you an idea of how the city prepared to war in renaissance times. In that case check the flag throwers too.
As mentioned, check out Museo Stibbert (which I never visited, despite being Florentinian).
Check out Siena too. Siena won against Florence in the 1261 battle of Montaperti, no idea if the battlefield is worth a visit. Nearby Monteriggioni shows how small medieval cities were fortified fortified.
In addition to the US war cemetery, there's also a Commonwealth one in the South and a German one on the mountains to the North (more or less along the Gothic line mentioned above).
Italy's oldest, I think, wargame club is the one in Florence, AFBIS. It is quite small though. You can find them on Facebook: link . There's some very knowledgeable and friendly guys there who speak English. They also have quite an impressive military, history and wargame related library, though most titles are in Italian.
This of course in addition to all general tourist places (that is, less militarily interesting), which you can find plenty of info on everywhere.
I'll be there in a couple of weeks and again in May, if you happen to be there then. For any question, drop me an e-mail at leopejo on hot mail.
This question has been asked other times too in the Wargaming in Italy section of TMP, you might find other useful answers there.
P.S. I don't get troopwo's last line.