Piazzale Michelangelo is no longer free parking in Florence and the number of open pay spaces is limited compared to the past. There is parking (not free) at Porta Romana and if there is space, you can park there and take the bus into the center or walk up Via Romana.
I would pick up the car at FCO which has the advantage of lots of cars usually and also not requiring you to drive in the center. Note that Italy is not the US and quick pick up of rental cars is not the norm. Expect it to take 2 hours and be pleasantly surprised if it is faster. I would go in about an hour before your reservation and hope they have a car and can get it all processed in that hour. It often takes much longer.
I might not do Florence -- it really takes several days to explore even the highlights of its Renaissance artistic heritage; it does not repay a day trip. Save it for next time and focus on Tuscany outside of its capitol. I would definitely give a day to Siena (lots of parking garages just outside walls of historic center) and visit the gardens of la Foce
https://janettravels.wordpress.com/category/tuscany/
Montepulciano, Monte Oliveto Maggiore or Sant Antimo, Lucca and of course Volterra which is near SG. Volterra has a wonderful Etruscan museum. We were once trapped inside for 4 hours by an incredible thunder/rain storm that took out the lights. So we actually looked at the hundreds of Etruscan burial boxes which are truly fascinating. Everything we know about the Etruscan way of life comes from the carvings on those boxes. Great museum.
South towards Rome there are some wonderful sites. We enjoyed Sovana and the Etruscan tombs in the hillsides nearby. We picked up a map in town and drove there. Another thing I think is a don't miss is the hot spring waterfalls of Saturnia
http://www.theguardian.com/travel/2010/may/01/italy-tuscany-natural-hot-springs
A couple we ran into at an Etruscan archeological site told us about them and we drove down and enjoyed soaking; no changing rooms -- we just changed by the car. Take turns so someone is watching the valuables. You will smell like sulphur for days, but it is worth it -- one of life's great moments lolling in the hot water gazing out over the countryside