Tuscany

Tuscany is one of those places that is best seen by a rental car. Roads are well paved and marked, however they can be a little crooked. Many of the great sights can be off the beaten path where there are no rail lines.
Many people consider the area to the south of Florence to be a centralized location for Tuscany. I especially like the countryside outside of San Gimignano, Siena and Poggibonsi to stay--45 minutes or so south of Florence.
Virtually every farm in Tuscany has apartments and rooms for rent as a secondary source of income.
You might want to get yourself a road map of Italy to give you a good perspective on the locations of the cities you want to visit. They are available online and in quality bookstores.

I agree with David that a car is very convenient in Tuscany. I think it depends on where you want to go. If you are interested in Florence, Cinque Terre, Pisa and perhaps a few of the surrounding areas such as the Garfagnana (reachable by bus), Lucca would be a good choice. If you're also interested in visiting other Tuscan towns such as Volterra, San Gimignano, Siena (also reachable by bus from Florence), and the Chianti region, having a car is ideal. As you say, you could rent a car for a day to visit some of the other towns not so convenient by train. I spent two weeks in Lucca a few years ago and loved it.

Enjoy your trip!

DD

Lucca is lovely but flat and it may not deliver that “heart of Tuscany” experience. I agree with the others: head a bit south. You might stay in Siena or Florence for 3 nights to experience that part of the region then head to Montalcino by car and experience the Val d’Orcia and some stunning hill towns.

From Siena you can daytrip to Volterra and San Gimignano although Lucca is a bit far, IMO, for a day trip. From MontalcIno there abbeys, Montepulciano. Pienza, San Quirico, and many more. But a car is essential for the most efficient travel.

Tell us what you plan to see and do.

Like you, we were time constrained. We also wanted one location. This was an anniversary trip (50 years since our first kiss in Florence) so seeing Florence was a must, though we wanted to spend most of our time exploring Hilton towns.

We stayed just outside Greve, which was great. It is on a main bus line to Florence, about 40 minutes away. I knew I didn’t want to drive in Florence. The rest of the time we drove around. We visited Siena, San Gimignano, Volterra and Radda. We also went south to the Monte Oliveto Abbey.

We had a great time. Staying put in one location worked out well.

Trains in Tuscany are limited. If you are willing to take buses, you'll do much better. However, Florence and (to a lesser extent) Siena are the bus hubs; from other cities, you will probably have to make changes to get to some places. Note also that bus service is quite curtailed on Sundays.

If you really don't want to rent a car, you should figure out what places in Tuscany you want to go, and investigate the transit options, before picking a home base. Rome2Rio can be a good start to see bus companies, but do be sure to check the website of the company itself for accurate information. It is good for identifying snags (like a place that will take 3 buses to reach, but the buses aren't coordinated, so can't be seen as a day trip). https://www.rome2rio.com/

If are not going to rent a car, you should stay in Florence and/or Siena. The most famous Tuscan towns can be easily reached by either.
If you rent a car, that would be a more efficient way to see Tuscany. In that case you will have more options on where to stay. My suggestion would be to find a villa/farmhouse/hotel/apartment in the countryside or in a small town. Anywhere between Florence and Siena would work. I’d prefer somewhere in between not too far from the main freeway connecting the two cities.
Lucca is an easy day trip from Florence, but it’s not central to the rest of Tuscany, therefore not ideal to visit anything but the western side of Tuscany.