Cinque Terre and Positano/Amalfi are NOT the same experience. CT is in a national park. You can hike, take a 5 minute train ride, or, in summer, take a ferry between the 5 villages. The Amalfi coast is primarily bus or taxi travel. Day 1, flight into Rome, morning arrival. Train to La Spezia. Local train to either Vernazza or Monterosso. Dinner on the harbor, or the beach, depending on destination. Next morning, train to Riomaggiore (5 minutes), then walk to Manarolo. Train to Corniglia (5 minutes), hike the trail to Vernazza, train to Monterosso. A nice seafood risotto at one of the beach restaurants. Day 3, train from La Spezia to Florence. Dinner in the shadow of the Duomo. Morning, Accademia. Afternoon, Uffizi. Evening Steak Balsamico at Aqua al Due. Rent a car here for drive to Volterra, then to stay in Siena. Next day, drive from Siena to Montepulciano, Montalcino, San Gimignano and whatever else you have time for. Stay in Siena. Drive car south to drop off at FCO Rome. 4 nights in Rome (you could stay 10 nights and not see it all). From Rome, train to Sorrento, for 3 nights. Day trips to Positano and the Amalfi Coast (primarily bus or taxi). Train to Naples for last two nights. Pick your favorite Roman site. Fly home from Naples.
2 nights CT, 2 nights Florence, 2 nights Siena, 4 nights Rome, 3 nights Sorrento, 2 nights Naples.