October is a great month to be there! You have 9 nights, right?
Assuming you arrive early enough (i.e., before say 15:00), I would proceed immediately to Florence for 5 nights. There are some direct trains from FCO, otherwise, a change at Termini is easy. Five nights gives you four full days, so perhaps two days for Florence and two for day trips. I would stick to Siena and Lucca. Venice is too far for a day trip and the Cinque Terre requires at least two nights to make it worthwhile, IMO. Pisa is a snooze but can be combined with Lucca.
After Florence, back to Rome for your remaining nights (4?) before departure. No car required -- nor recommended -- for this itinerary.
If you really want to add the CT or Venice, I would go Florence 3 nights, Venice or the Cinque Terre 2 nights, Rome 4 nights, in that order.