Since you're planning on getting a car, I would suggest starting in Florence, seeing the city and then also doing a few easy day trips from Florence with train and only after renting a car to head out to the countryside to your next base from which to explore.
The easiest thing really is to rent from Florence, if you're worried about driving in the center you can take a taxi to the airport and rent from there. The airport is close to the highway so you don't have to drive back into Florence at all to head out of the city.
So a suggestion could be:
Florence - 3-4 days, one of which can be a whole day trip out divided between Pisa (in the am) and Lucca (in the afternoon).
Rent car and drive out to the countryside.... here you can decide whether you want one base for the rest of your time or two, if you don't mind packing up again and moving to another spot.
Usually I would suggest one central base, in the area between Florence and Siena or closer to Siena.
Then you could do spokes of the wheel and visit in a different direction each day.
Chianti - 1 day
San Gimignano and Volterra - 1 day
Val d'Orcia - 1 day
Arezzo and Cortona - 1 day
If you want to head further away, maybe a second base would not be a bad idea and head to
Maremma, in southern Tuscany along the coast. There I would suggest a day exploring the towns made in tuff rock, such as
Pitigliano,
Sorano and Sovana and another day at the
hot springs of Saturnia.
Another day spend it at the beach - I particularly like the sandy beaches at the north end of the Maremma Park of the Uccellina.