Proprietors of the strictly made Chianti Classico can be found in abundance, although the Castello di Vicchiomaggio and Castello di Verrazzano are stand-outs for ambiance and reputation. Tastings can be had in town at Le Cantine on the main square, which has an underground cellar over 500 years old. If you can cobble your own wine tour together, you'll find that it can be a pretty cheap way to spend your day (most local tastings are free) in a gorgeous, romanticized world away from the big cities of Italy.
Then again, we'd be remiss to overlook the more active aspects of a potential adventure in Greve. Try the day long hike to Monte San Michele from the Palazzo della Torre, the views are wonderful and the food at the mountain restaurant is quality while the more moderate walks to Montefioralle and Panzano beyond are an excellent way to explore the Chianti region a bit.
Note:The Chianti Classico Wine Festival is held every year in Greve on the second weekend of September.