Originally pairing Rome w/ Florence and Tuscany but have been to Rome many times. Do not want a rental car so relying on train and bus. Museums not our principal interest! Arriving and departing from Rome . Oct trip .Thanks!
Consider Bologna. It's a pleasant town with few tourists, with good restaurants and excellent rail connections, making it easy to day trip to charming, interesting places like - Modena, Ferrara, Padua, Parma, Ravenna (a bit farther but worth it). It's about 30 minutes farther on the fast train from Rome, so you could start there and end in Florence.
What are your interests? If not museums, why base in Florence? We often make our base in a town with a train station near a rail hub town rather than in the hub itself. For instance, you could stay in Parma rather than Bologna, or Pistoia rather than Florence. But it would help to know what you are interested in.