You can book the winneries by yourselves. Please keep in mind that much of the countryside will not be easily reachable with public transportation.
In theory, 7 days would be enough for what you are planning but being without a car and with a limited budget, you are looking at ALOT of walking and biking to get around. Now that i warned you:
Consider staying in florence for the time you wish to dedicate to seeing it, then move to either Siena or San Gimignano. Siena might be less expensive overall and you would proabably have more transportation options. San Gimignano is basicaly a "small" town that is overcrowded with winneries, and if you are coming for the wine, that is your paradise.
I would consider booking a wine tour for 1 day, as reaching the farms on bike and getting home after more than 1 small glass of Chianti doesn't sound like a good idea to me. Ofcourse you might have a different idea.
I would also rent a cheap car for 1 day and do a tour of the medieval cities.
One more thing. The Montepulciano area is home to some of Italies most expensive wines, especially the Brunello. Thus, any wine activities related to them will be expensive.