Tuscany

Bologna is a great base for day-trips and a very nice city besides, but it's in Emilia-Romagna rather than Tuscany, so it probably will not be a particularly central location for you if you specifically want to see Tuscan towns.

A lot of the places most often spoken of in Tuscany do not have train service, or the train station is down at the bottom of the hill, so you'll almost certainly need to use buses for some destinations, and you'll typically find it challenging enough to get to two places in a day; three is usually not going to work out, no matter how little time you might need to see each one, because of the bus schedules.

I think the first thing you need is to get hold of a comprehensive guidebook that covers at least Tuscany. It would probably be smart to have wider coverage than that, because you may indeed find you are intrigued by places that can be visited from Bologna. (I especially loved Ravenna.) Once you have a target list (don't worry if it's too long--everyone's list is too long), you can look at transportation links and figure out your best option for a base or bases.

The website Rome2Rio.com may be of help in the early stages of planning. You cannot trust its travel times, bus/train frequencies or fares, but it has proven useful in indicating what mode(s) of transportation are possible between two places. It will, for example, show you when rail service alone will not do the trick. It will suggest the logical place to change buses or to switch from a train to a bus if that is necessary.

For trips that can be completed totally by rail, I find the Deutsche Bahn website the easiest one to use for research, though you will ultimately need to go elsewhere if you need or want to buy tickets online before your trip.

All that said, Florence is generally considered the best single base for seeing Tuscany without a car. Still, to make an optimum decision you need to know what towns you want to see.

You can cover alot by bus from Siena. That is what we did.

Florence is probably the best base if you are traveling by train. There are some great cities in Tuscany that you can reach by train. As suggested, research the cities in Tuscany and decide what interests you. Here is a website which will help. https://www.discovertuscany.com/

In my personal ranking, Florence is the best base to visit certain famous parts of Tuscany without a car.
Siena would come as a close second choice, as it is also great as a base to visit other famous parts of Tuscany.
Arezzo or Grosseto would be competing for third place, depending on which towns you plan to visit.
Trains are good to visit provincial capital cities from Florence.
All others are easier by bus.

Been to Bologna, it's OK, but pales compared to Florence.

You can do a day trip to several places from Florence. I see that you have been there already.

What places are you interested in visiting?

Siena is wonderful and would make a great base for exploring by bus.

Another option may be Lucca. It is a charming walled city with great train connections to Florence, Pisa, and the beautiful Garfagnana mountain region in the Northwest of Tuscany. If you're being ambitious, you could even hit the Cinque Terre.

I stayed in Lucca for two weeks while attending a language school and loved it. I got to explore a lot of the region when not in class and it was super convenient without a car.

Instead of a base to visit many towns I'd be inclined to stay in a few towns and really see the place, especially at night. Siena has bus connections to Montepulciano which would be a good spot for a night or two. It makes more sense to have a base if you are driving. For day tours, Florence would be good.

I would stay in Florence, as it is the best hub for the places you want to see, not to mention being a wonderful place to spend your days/evenings off from traveling around Tuscany. From Florence, Bologna is only 35 minutes via train, Siena is 1 hr, Lucca is a little over an hour (you can also hit Pisa, Prato & Pistoia on this route), as is Cortona (you can also hit Arezzo on this route).

Many would say that if you truly want to explore Tuscany, a car is almost a necessity, as most of the wine country and hill towns are not accessible by train, and buses can be time-consuming with stops & transfers, while the schedules can sometimes abbreviate your day. However, from Florence there are numerous bus tours that will take you into wine country if that interests you, or to hill towns like San Gimignano & Siena. You can also hire private car tours which will take you to some of the less-touristy places like Volterra or some of the smaller towns in Chianti, but that will be expensive.

Thank you all for your suggestions, especially the Discover Tuscany site. This travel forum and the many questions and answers by the readers has been most helpful.

I believe there are bus tours that include Volterra.