Best long-term stay location, Italy

Since there is a fast train to Rome via Florence, you don't necessarily have to stay in between the two, so I would look along the regional train lines in Tuscany. On the trenitalia website you can zoom in on those:
http://www.viaggiatreno.it/viaggiatrenonew/index.jsp

List all the places you want to visit and we can better advise.

South of Rome is kind of too broad a description. Where exactly? Calabria and Sicily are also south of Rome, but it takes 8 hours by train from Rome.

Florence and Rome are only 90 min apart, but only because the two cities are connected by a high speed train, otherwise even they are distant. They are almost 200 miles apart and if you drive 3.5 hours aren’t enough. If you choose any place in between you will need a long time to either.

@Roberto, the OP wrote "south to Rome," not "south of Rome."

Good suggestion above, look for a town in Tuscany or Umbria with train service between Florence and Rome, or Pisa and Rome, or Florence and Pisa.

@Bonnie, I don't know that area well but I'd think you might want a car to reach some of the best smaller places and give you flexibility.

Check out Lucca, which is on the main line between Pisa and Florence and when you train to either of those places, you have access to anywhere- north or south.

It's really important to look at train times from any potential base to the places you want to go. If traveling to smaller towns you will not be on a fast train. Even Siena to Rome is likely to take over 3 hours. You can easily check travel times on the Deutsche Bahn website.

Without a car, stay in Florence. Choose a pedestrianized neighborhood in the Oltrarno (south of river Arno), for example San Frediano, an easy walk to the station, and you will feel like in a smaller village. If you stay out of the main tourist drags, Florence doesn’t feel like a big city.
Siena is a good second base to visit the south of Siena province.