Cities/Towns near Roma?

The high speed trains generally make no stops between Rome and Florence, with very few exceptions.
Along the old standard speed railway between Rome and Florence, the obvious choices are Orvieto, Castiglion del Lago, Cortona, Arezzo, which are all served by train stations (Cortona’s is in Camucia, at the bottom of the hill). Castiglion del Lago is kind of small, but all others can keep you busy for the best part of the day.
If you take a slight detour, you could consider some choices in Umbria, such as: Spoleto, Foligno, Spello, Assisi, Perugia, to name a few.
It also depends on how many nights you want to spend along the way. It wouldn’t be bad to spend a night or two in Orvieto, and a few nights in Arezzo. Arezzo can be used as a base to visit several nice towns such as Cortona, Anghiari, Castiglion del Lago, and even Poppi, or Lucignano, or Loro Ciuffenna, and others too.

I did it in reverse, Florence to Assisi to Orvieto to FCO via Termini.

"I will not have a car and will travel by train only."

Don't forget that for many places in Tuscany and Umbria, buses work better than trains (particularly true if a town has no train service at all!)

Agreeing with Harold. Buses are often a better deal as they are more likely to drop you off near the historic sites you want to see, rather than the train station, which might be a good ways away. Pisa and Assisi are good examples of that.

We traveled by local train to Assisi and Perugia and then took a bus from Perugia to Siena and the into Florence. The journeys were just fine.