Yes, without a car I would choose Siena as a base for day trips to several hill towns in the province, such as:
Monteriggioni
Colle Val D'Elsa
San Gimignano
Barberino Val D'Elsa
Certaldo
Montepulciano
Pienza
Buonconvento
Possibly the Chianti hills, although those are better by private driver.
It takes longer by bus however, and bus service is really scant on weekends, therefore you would need more days available to cover all the towns you might want to see.
Florence is a good base for towns/small cities connected by rail such as:
Arezzo,
Pistoia
Montecatini Terme
Montelupo
San Miniato
Pisa and Lucca
Even Orvieto, although that one would be easier from Rome.
Assisi and the rest of Umbria is farther, and not easily reachable from Siena without a car, and the trip by train from Florence is kind of too long for a day trip.
Civita di Bagnoregio is out of the way and a car is almost a must to get there. But, it is of any help to you, in spite of the hype by Rick Steves, it's not really a big deal. Most of my friends who live in Florence have never seen it or even heard of it. It is actually more famous among Americans (probably because of Rick Steves) than Italians, just like Castelrotto (Kastelruth) that few Italians visit or are aware of, even though they may have gone to the Dolomites more than once.