How best to visit Tuscan towns without a car

Look into taking buses as well as trains. We had a car in Tuscany and didn't have parking problems, but we traveled in December. I'd pick some towns and then look up train and bus schedules. Compare that to the time it took to find parking. For instance, I looked up Florence to Radda by bus and it was ~3 hours, but 1.5 hours from Siena. Driving from Florence was an hour and 40 minutes from Siena.

For fun, I looked up buses from Siena -->San Gignano--> Volterra-->Siena. You can leave Siena at 8:30 and get to San Gimignano at 9:30. Then take the 1:00 pm bus to Volterra and arrive at 1:40. Returning from Volterra to Siena, it's 4 hours back home at either 5:00 pm or 7:00, arriving at 9:00 or 11:00.

Then by train to use Siena as a base to see other Tuscan towns. What
is the best way to get to Montepulciano, Montalcino, San Gimignano,
etc.?

With a car. There isn't good train service to any of the towns you listed.
As explained above the buses are infrequent and slow with spotty service on weekends.

Siena and Lucca are easy to visit by train from Florence (or bus to Siena).

toursbyroberto.com might have what you are looking for.

First of all, you need to do a little more research and see what towns interest you. You can see a lot of towns throughout Tuscany using public transportation, it just takes a little longer than if you drove yourself. As for visiting vineyards and wineries, however, I would think about taking a tour.

We are in Italy now and I had the same question. Last time we were here we rented a vehicle which is the easiest way to travel to some of the smaller towns. However, this time, we wanted to home base in Florence for 6 nights and take day trips. We were with another couple and took a tour with a private guide Fiorenza Bettini (Flo) to Val D’Orcia (Montepulciano and Pienza with a stop at a little village with an amazing view and food. We chose Flo because we stayed with her mother at her mother’s B&B in 2018 and then Flo held Zoom classes throughout the pandemic. We were so pleased with the trip. She curated everything to our liking and we had the most fantastic day. She had a driver pick us up at our hotel in Florence and she arranged wine tastings, lunch, and tours. She made sure that we had time for photos and while the day went longer than we contracted, did not hurry us or charge us more for the extra time. It was truly a memorable experience.

If you want to do something with a shared tour, you can also look at RS recommended guide Roberto as an option.

RS goes into great detail describing how to travel to the better-known Tuscan hill towns using public transportation in his Italy guidebooks. Bus is required in many cases. There may also be tours offered from both Florence and Siena.

Any recommendations for tours from Florence and Siena? Food, wine, cooking, vineyards with lunch, etc?
Thanks for all the advice!

ToursbyRoberto.com

My family and I did not have a good experience with Tours by Roberto. We very may well have caught him on a bad day, but it was an unpleasant experience with lots of snide remarks to polite questions. I personally would look for an alternate company.