21 days - I love it!!
Buongiorno,
I suggest you divide Tuscany into three even four sections.
Hold off on getting the car right away and start your time in the city center - take a few days to enjoy Florence/Museums/Monuments and Churches. This will give you a chance to get over a bit of jet lag and travel fatigue before getting behind the wheel. And it will give you lots of time to get organized and get the lay of the land.
Areas that I would definitely check out:
Chianti
San Gimignano (Volterra!! and the surrounding countryside is awesome)
Val d'Orcia - take the time to go south and look for Pitigliano and the area, not only great wines but some beautiful landscapes, towns, and recipes.
If you really want a bit off the beaten track then
Arezzo / Cortona & Casentino
All of these areas have dedicated sections on the website, and I encourage you to have a read through the many articles. The authors are all on-site in Tuscany, and we write what we know about ... so you will get some good solid local info.
As for a cooking lesson - there are so many places that offer that now with the B&B or agriturismo service, you really should plan on more than one or two.
Browse through this link for some ideas:
http://www.tuscanyaccommodation.com/for-foodies/
some of my favorites:
https://www.tuscanyaccommodation.com/borgo-argenina.html
https://www.tuscanyaccommodation.com/casolare-di-libbiano.html
Buon Viaggio,
Donna Denise