You're going during a great time! There are plenty of festivals throughout Tuscany then. Do a Google search.
I think where we stayed would be just right for you. We stayed at Castillo di Verrazzano (http://www.verrazzano.com/), which is just outside Greve. This winery produces some exquisite Chianti Classico, as well as a tasty Super Tuscan. In addition, they make olive oil, wild boar salami and an amazing traditional aged balsamic vinegar.
There is a cozy agriturismo on the property with 8 rooms (I think) available. The rooms are extremely clean and comfortable. The furniture in our room looked more like pieces you'd see when you stayed with your aunt instead of Howard Johnson motel furniture. That added to the charm. Check the website, but I remember the rooms running around 90 euro including a breakfast to die for.
You can take a fascinating tour of the winery. I highly recommend you finish the tour with a food and wine tasting. It starts with Prosecco paired with an appetizer, then goes on to their Chianti Classico, Chianti Classico Reserva, their Super Tuscan and finishes with Vin Santo and Biscotti. Along the way they pair their wines with wonderful meat and pasta dishes. Right before dessert, you get to sample their balsamic vinegar.
Greve is just 5-10 minutes down the road. It is a charming little village. The good news is there is no ZTL and there's plenty of free public parking available. There's a great Chianti festival at the end of September, just when you'll be there! Be sure to have dinner at La Cantina. We ate there twice and both times we were the only non locals there. Alessandro, the owner, bends over backwards to make you feel at home. You can take cooking lessons from his mama, Mama Lorena (http://www.pizzerialacantina.it/cookery-school.html).
Verrazzano turned out to be a perfect home base for us. We were able to take trips to Florence, Siena, San Gimignano, Volterra and Montepulciano, as well as several small hilltop walled villages like Radda. The scenic back roads are a joy to explore.
Buon viaggio!