Chianti is a great base for 5-6 days, with rental car to move around
Sounds like a wonderful birthday gift!!
Chianti would be a great base. I'd recommend renting a car in Florence, then driving out to Chianti for your 5-6 days. Chianti is very central and from there you can both enjoy the surrounding countryside as well as day trips to other parts of Tuscany. You can read more about the car rental from Florence here:
http://www.visitflorence.com/florence-car-rental/
You can then visit Florence as well on a day trip, drive around Chianti (read more about the towns here:
http://www.chianti.com/ ), visit San Gimignano, Siena, Volterra, Monteriggioni, Pisa... there are many choices.
Here is an itinerary for 5 days in Tuscany:
http://www.discovertuscany.com/itineraries-in-tuscany/five-days-in-tuscany.html
As for B&Bs, I'd recommend the following:
http://www.tuscanyaccommodation.com/palazzo-malaspina.html - small b&b in a small medieval walled village close to the Firenze-Siena roadway, making it ideal for day trips. Also just great place with delicious homemade cakes and jams for breakfast made by Maria. She can help or direct you to cooking classes nearby.
http://www.tuscanyaccommodation.com/casolare-di-libbiano.html - great small b&b in the countryside closer to San Gimignano in farmhouse and large garden. offers cooking classes directly at the b&b.
http://www.tuscanyaccommodation.com/tenuta-quadrifoglio.html - has both apartments and rooms, offers cooking classes too I believe.
http://www.tuscanyaccommodation.com/relais-poggio-borgoni.html - B&B on a farmhouse also provides win tastings, olive oil and has own restaurant. Very centrally located.
For additional accommodation in Chianti, take a look here:
http://www.tuscanyaccommodation.com/chianti-accommodation/
http://www.tuscanyaccommodation.com/chianti-bed-and-breakfasts/
Hope these ideas are useful, let me know if you have any other questions!