We spent 8 days in Tuscany last December. We went everywhere you mentioned except Pisa that got left off the list because of time restraints.
We stayed in Greve, which is centrally located (20 km south of Florence) and did day trips. I hope our experience will help you have a great trip.
1) Drive or use public transportation? We did a little of both. There's a ton of back road scenery that hard or darn near impossible to see using public transportation. We enjoyed the freedom of exploring on our own time schedule. If you are a careful driver, you'll do fine. Driving in Tuscany is not much different than driving in the US. Obey the speed limits. Learn the signage before you go (it's not that hard). Take a Garmin with an SD card of Italy. I spent an hour at home before we left entering and saving every address, including parking lots, that we planned to go to. By the end of the trip, I knew the region so well that I didn't have to rely on it to get around. I think the dreaded ZTL problem is overblown. The zones are well marked and you have to not be paying attention to make a mistake (although I almost did in Siena).
The one time we took public transportation was from Greve to Florence. Buses run every 10 minutes or so since many live in the country and commute into Florence. We were the only non-natives on the bus and had a great time! It's only 4 euro each way. You can catch the 6:45 bus from Greve and be in the heart of Florence by a little after 8:00.
2) Florence - Give yourself 2 days to visit. I wouldn't recommend it as a base because you'll chew up a lot of time getting in and out of the city. We only gave ourselves one day in Florence. I regret we didn't spend at least two. We used Walks of Italy to do a morning walking tour that included spending an hour at the Academie to see David and other works of Michelangelo. We also visited the Duomo We then did the afternoon tour of the Uffizi. They provide your entrance tickets and you skip the lines. I wish we had spent another day to visit some of the other museums and churches. It would be worth the expense to have rented a room in town and stayed overnight.
3) Volterra and San Gimignano - We did both in a long day trip. We spent the morning at San Gimignano, had lunch then drove to Volterra. Both have large parking lots outside the towns and you can leisurely walk from one end to the other in an hour or so. Make Volterra your afternoon trip. Walk just outside the city walls and watch the spectacular sunset, then have a nice dinner before driving back.
4) Siena - You can do Siena in a long day, but two would be better. It's not as jam packed with museums and sites as Florence, but you could easily fill two days at a more relaxed pace.
5) Pisa and Lucca - We had them on our itinerary, but decided we would be too pushed to fit them in. We had been to Pisa before and there's not much more than the tower and baptistery. Instead we took a day to ramble south from Greve, stopping at the charming small town of Radda and ending the day at the Abbey of Monte Oliveto Maggiore. The abbey was a highlight of the trip. The evening vespers are in Gregorian Chant. Besides a lady from a nearby town, we were the only ones there other than the monks. It's a gorgeous, calming service in a spectacular church. It's also close to Montalchino and Montepulchiano. You could do Radda in the morning, catch Montalchino for lunch and the afternoon, then drive down to the abbey in time for the vespers at 7:00 (check the time).
6) Final notes. We scheduled in a down day the first full day we were there. We stayed at Castillo di Verrazano so we had a leisurely breakfast, then took the wine tour, followed by a wine and food pairing that took three hours! The owner kept bringing over samples of new vintages for us to taste and provide feedback.
We spent three days in Amsterdam in July and had a great time! Have fun!