Welcome to our forum! That sounds like a marvelous plan! 3 weeks will give you a good chance to discover the small villages off most people's radar and areas that are cool to explore. If you will have a car, it will make it much easier to move around and spend your time wisely, including moving to a different area every 4 days. You will have to see and
Look into staying in Maremma, in the southern part of Tuscany. The small villages of Pitigliano, Sorano and Sovana, Saturnia... then along the coast Capalbio, Orbetello and Talamone. Wine country around Scansano, and a bit north Montecucco.
Another area: Valdorcia. Towns of San Quirico, Montalcino, Pienza, Montepulciano, Buonconvento. Wine areas of Brunello and Rossi di Montepulciano, as well as Valdorcia.
Move on to Arezzo and explore Cortona and Sansepolcro, Anghiari.
Move on to Chianti to explore Castellina in Chianti, Radda, Gaiole, Panzano, San Donato in Poggio, Greve. Make sure to visit Siena either while here or in Valdorcia.
Then there is Florence. I would say a minimum of 3 days.
Last, you could head back toward the coast between Pisa and Lucca. There is a small hill range called Monte Pisano in the middle that is off the radar of most people but has neat villages: Vicopisano, the Certosa Abbey, the spa town of San Giuliano Terme, the Pieve di Compito and nearby camellia garden. This is lovely for people who love the outdoors, as there is chance to do hiking here too. From here you could head to the beach: either to Tirrenia to be close to Pisa or further north toward Viareggio. Make sure to include Pisa and Lucca, at the very least half a day for Pisa and a day for Lucca.
I think I've given you lots of areas to consider -- now there is wine tasting in all of these areas, so with sightseeing, drinking and eating, I think you will easily fill up those 21 days! Plan to have Florence either at the very start or end so that you can use the car the rest of the time.
Take a look at the entire site - here is just an article for Pitigliano: https://www.discovertuscany.com/maremma/pitigliano-highlights.html
There are similar articles for most of all the places I've mentioned above. So get in, get lost and make notes. Let me know if you need more info or ideas as you make your plans and have fun!