Hello Anna3,
I like the idea of your plan - the opposite of people planning to visit 13 cities in 12 days - but I would hesitate to ever have a car in Florence. There is obviously no need for one in the city but unless you arrange to stay in a place that provides parking and a clear path into and out of the city I think the pressure of a driving a car into even the edges for Florence would probably outweigh the issues with changing home bases.
I would stay in Florence and spend a week or so - exploring the city and taking day trips to places you want to visit that are well served by train and bus and do not require a car. These include Lucca, Pisa, Arezzo, Bologna and anything else that might interest you. Then I would rent a car and then choose a town of interest and stay there and drive to explore Tuscany further by car.
Many of the famous Tuscan towns are a ways away from Florence and staying away from the city will shorten your drive times and maximize your country driving experience versus your city driving experience which is generally less pleasant.
Places like Pienza and Montalcino are realistically 2 hours by car from Florence but very close to each other and about an hour from Siena. So in my mind it makes them a day trip from Siena but a really long - probably too long - drive from Florence.
I would make a list of the places you want to visit in the Tuscan countryside and see if there is somewhere that is a larger town that is central-ish to these places and think about staying there for a few days.
Also remember that it is possible to return the car somewhere else if you arrange it if you are well south of Florence when you are done rather than driving back north to the city to return the car and then training back south. Orvieto and Chiusi are popular spots for this since they are close to the train line that leads to Rome and onto Naples.
Sorry to give you more options when it sounds like you're trying to keep it simple but small town Tuscany is one of those regions that require a car and some planning. But don't let the hassle discourage you - the region is as beautiful as advertised and more so.
Have a great trip,
=Tod