Staying In Tuscan Hills Towns

You may be able to take a train to San Gimignano or Certaldo (according to Rome2Rio.com) from Florence, but to Greve you'd have to take a bus. Then, of course, getting to Florence or Siena for sightseeing woudl require the same transportation. You can do some basic research on options and transfer times at R2R.

However, if you want to see a lot of the small Tuscan towns, a car is very convenient. Take care though as driving INTO Florence and Siena is problematic. You have to park peripherally and watch out for the dreaded ZTLs.

Rent a car, you will enjoy the freedom.

Driving to Siena is easy, parking isn't bad in their parking lots. We used Santa Caterina parking lot which is pretty easy access to town and easy to get in and out of unlike some parcheggio's.

San Gimignano is a bit out of the way except for Certaldo and Volterra, but you can do day trips to Siena easily. We stayed in Montecatini Terme once and made easy trips to all the towns you mentioned plus 5 Terre, Lucca, Pisa, Vinci, etc. So maybe somewhere like that would be more central to your day trips. It is easy to get to Florence by train from any of the places along the Arno or along E76 autostrada.

Getting out of the Florence airport by car is easier than renting a car in town, you have fast access to the autostrada.

If you do stay in Greve do go to Panzano and see Dario the Butcher, he is a character. Search him on You Tube, very funny guy. Driving in that region is a bit more difficult as it is hilly and curvy roads abound.

Have you actually rented a villa in a town center? Most are located outside of town. I can't imagine staying in a villa without a car, or actually, staying in Tuscany, Umbria, Poemonte, etc without one.
Greve is not a hill town, bit a nice larger town, flat in town center. Do you only want to stay in a hill town or just a nice Tuscan town?

The reality of renting a Tuscan Villa might be different than the idea. I like to go out to dinner and walk home therefore I prefer to stay in the hilltowns. If I had a family a villa would be nice but in either case a car gives you freedom to do whatever you like. A GPS makes it much less stressful.
Get the car when you leave Florence and/or Siena. Last time we went from Florence to Siena by train and stayed near the station outside the walls (you don't want to drive inside the walls of most hilltowns without a permit from a hotel). It made day trips and departing from Siena easy.

not sure if this is too close to Florence for you but this is the villa we are staying at-- renting a cottage. We have the option of having them help us secure transportation to the Villa. This place is in the Chianti region about a 15 minute drive to Florence. 5 min. walk to the town of Imprunetta. The bus stops by the entrance to the villa every 30 minutes. We are going this September. We plan on taking the Train from Venice to Florence and then getting our transportation to the Villa. We like that we have the option of renting a car if we want to explore further in Tuscany but we already have set dates and will be heading to Rome after 4 full days. http://www.relaisvillalolmo.com

Although virtually everyplace in Italy is accessible by public transportation of some sort, for the type of trip you are envisioning (I.e. the small town of Tuscany) a rental car is much more convenient.

And now to your questions. I rent cars in Florence every year including now (I'm in Tuscany now). Generally I rent a car in the city of Florence a few days after arriving. but this time I picked up at the airport upon landing (taxi drivers were on strike in Florence, I think against Uber). The rental car center is across the freeway from the terminal so you need to take the rental car shuttle bus. It's a 3 min ride but don't be surprised if you have to wait half hour for the shuttle bus. After you pick up the car just drive to your villa. Ask for directions at the rental center but if you have reservations study the route on Google maps ahead of time.
From your villa, wherever it is, you can drive to Florence or Siena. Just park in one of the parking lots outside the city walls. For Florence use the parking lot called Parcheggio Oltrarno on Piazza Della Calza. For Siena use the Parcheggio Fortezza near the stadium. Use google maps for directions.

San Gimignano is not served by train. What Rome2rio shows is a train to Certaldo and then once-daily bus from there to San Gimignano (but there are more buses from Poggibonsi). I, too, would rent a car for more freedom.

We got home yesterday. We rented a car at the Florence airport and drove to San Gimignano, stayed for 3 nights venturing out to Siena and wine tasting. Drove to Orvieto and turned in the car at the train station there. Rented from Hurtz. Driving was actually not bad at all. I cannot stress enough that you should rent a hot spot with your car for navigation. Google maps saved us and we didn't have any wrong turns or navigating arguments. I did the driving (I drive faster) and my husband navigated.

All of our apartments had wifi advertised and none of them had wifi that actually worked well so that is why I am suggesting mobile hot spot over an in car navigation. You will not find a villa in a hill town, you can find very cool apartments at the top of long staircases. You cannot drive within many of these towns but there are parking lots outside. I vote for car. It worked beautifully for us.