With 2 weeks of time, you will be able to see plenty. And two bases would definitely be good, to allow you to explore each area without wasting too much time moving around.
You will, however, be limited in your choices of location and places to stay at when you have no car to move around on your own. The best solution is to stay in a town and not outside in the middle of nowhere so that you can easily catch a train or bus. For this reason, Florence and Siena make the best bases - with lots of trains and buses everywhere, they make it possible for you to use then as a base and do day trips out. Especially if you won't have a car and can have the opportunity to join group tours that will take you out to places you wouldn't reach easily otherwise.
For Florence, a good option would be to stay at
Villa Medicea di Lilliano:
https://www.discovertuscany.com/tuscany-accommodation/villa-medicea-di-lilliano.html
It gives you a taste of the countryside but being less than a mile away from the small town of Grassina where you can catch bus either into Florence (which is about 15 min away on the bus) or further into Chianti. They have great views of Florence from up there, too, which doesn't hurt
From Florence, you can catch buses and trains to do day trips, such as the ones suggested here:
http://www.visitflorence.com/itineraries-in-florence/top-day-trips-from-florence.html
Another two good options would be
Torre a Cona:
https://www.visitflorence.com/villas-near-florence/torre-a-cona.html which also has a bus passing nearby that brings you back to Florence, and
Viticcio just outside of Greve in Chianti:
http://www.chianti.com/chianti-farmhouses/fattoria-viticcio.html which is a short walk from the town. In Greve there are restaurants, cafes and buses back to Florence and other parts of nearby Chianti to also make it easy to get back to Florence.
All of these are wine estates so you can enjoy winery tours, wine tastings and cooking classes directly at the estate farm apartments.
If you want to explore more of
Chianti without a car, especially wine tastings, I suggest joining group tours out of Florence:
http://tours.discovertuscany.com/en/8357/Tuscany-tours/Wine-Tasting-and-Winery-Tours/d206-g6-c21
As far as a second base, I'd recommend
staying in Siena:
https://www.discovertuscany.com/siena/accommodation/
While it is a town, staying there and waking up in town makes you realize it has maintained a small town feel but with a lot of convenient restaurants and cafes nearby. You can also do day trips from here more easily, particularly into the
Val d'Orcia which is hard to reach without your own car. Take a look here for suggestions on possible tours you could find interesting to join in from Siena:
http://tours.discovertuscany.com/en/8357/Siena/d944-ttd
