The Gulf of Naples area (Naples, Pompeii, Sorrento, Capri island, Amalfi Coast) can easily be done without a car. Take a high speed to Naples, then a commuter train, called Circumvesuviana train, to Sorrento. Pompeii is between Naples and Sorrento. Sorrento is a good base to visit also the island of Carpi (by boat) or the Amalfi Coast (by bus or private driver or moped). Some people prefer to stay in the Amalfi coast, which is not as convenient logistically but quaint and romantic, so it’s up to you.
You can reach Puglia by train, but some of the locations are more efficiently visited by car, like the Garbanzo peninsula (Vieste, Pescichi, etc.). The towns along the Adriatic coast (Pollinmano, Lecco, Monopoly, Ostini, Oltrarno, etc.) are served decently by train, but as you go to the interior (Alberobero, Leccorotondo, etc.) buses may be more convenient. Overall I would consider renting a car for Puglia, for example picking it up in Bari. Then from Bari you can return to Rome via high speed train. Or you can fly home from Bari (with a layover in Rome or somewhere else).