You're getting good advice, but I would steer you slightly differently: The good news is that this is your first trip to some pretty magical places in which you can effortlessly and meaningfully fill the time you have without straying far. I spent a month in Firenze twenty year ago and only left the city limits twice (once to meet a friend flying into Rome and once to sit in the Piazza dei Miracoli in Pisa and share a panino with a cute girl) and never lacked for interesting things to do. So, I say focus your time seeing those three wonderful cities. If you want to see a winery, don't waste a lot of time getting out of town and visit something near Firenze, maybe with an organized tour (there's a guy who leads a caravan of old Fiat 500s, so you can drive a vintage Cinquecento while getting liquored up and not get lost: maybe not the best idea, but I'd do it).
The not so good news is that Tuscany is pretty sophisticated about wine, so they've focused on the world market successfully. You'll see more "world" wines than I'd like (Cabernet, Merlot, and even now Malbecs). You can still find wines from traditional grapes (Sangiovese and Vernaccia are the two big ones) and the 2009, 10, and 11 Chiantis are pretty good, especially the 2010. If you want to spend more, there are plenty of Brunellos and various Super Tuscans to try. The wine cellar at Enoteca Pinchiorri is said to stock >100,000 bottles and Cibreo will feed and wine you in fine fashion. Me, I'd stop at 'l Trippaio in San Frediano for a plate of lampredotto and a simple chianti (or Vino Nobile di Montepulciano)
Perhaps the best solution to the challenge you set was recommended by Baudelaire:
Il faut être toujours ivre. Tout est là: c'est l'unique question. You can do a lot worse than the wines from Tuscany, or maybe, if time permits, pass through Verona on your way to Venice and order a great Amarone.
Buon viaggio.