Oh you will already be staying in the heart of gorgeous landscapes with great views of Chianti!! On your way to Florence, Siena and San Gimignano you'll be getting more of the Chianti countryside with vineyards and olive groves, but if you want to get the rolling hills with cypresses you'll need to also add in a drive through the Val d'Orcia. Head to Siena, then on to Montalcino and to San Quirico d'Orcia, Pienza and Montepulciano. You'll get the landscapes you see in postcards from the region there.
If you also take the back road to Pisa and skip the FI-Pi-Li you'll get more gorgeous views, but also the faster road cuts through valleys that have views of hilltop towns off the road as you drive there.
As for wineries, they are all over the place. Are you interested in particularly beautiful locations or architecturally modern? There are small boutique wineries across the region, I would highly recommend you look into these three I've highlighted in a recent blog post:
The Best Wine Experiences in Chianti this Summer Tuscany is best experienced relaxing, with a view in front of you and a glass of wine in your hand. Across the entire region you can find “local Tuscan wines” from that part of Tuscany. While the Brunello di Montalcino is pretty well-known...
www.discovertuscany.com
Dianella is closest to Montaione but if you're going to Greve, I highly recommend Viticcio.
As for modern, everyone will tell you to go see Antinori and that isn't far from Montaione, it is a gorgeous place. If you love them, I'd suggest to go to Badia di Passignano for the gourmet meal with Antinori wines.