Rome-Siena-Florence will easily take up the 7-10 days. You won’t need a car. Trains are the way to go. If you wanted to take a day trip to say Chianti and Montalcino, Siena is probably the best location from what I remember. There is a Hertz location across the street from the train station in Siena. While Florence may be slightly closer, you run the risk of turning into a no-car zone and getting a fine in the mail.