If you want to go beyond the Rome-to-Amalfi stretch, I can highly recommend Puglia. I used public transportation, so I chose to stay in Lecce, a gorgeous baroque city that's great for wandering and is a sort of transportation hub for the surrounding small towns. Last summer there were very reasonably priced (55 euro) van trips from Lecce to tourist hot spots like Alberobello, coordinated by the tourist office but run by a private company. Usually that sort of thing is very expensive--likely over 100 euros for a day trip. The Lecce deal is transportation only, hitting (I think) 4 to 6 towns in one day. You cannot do that via public transportation in Italy. These aren't guided tours, so you need to have your own guidebook or information on your electronic device. The driver took us to a spot convenient to the highlights of each town, pointed us in the right direction, and told us when to be back at the meeting point. It was a great way to cover a lot of interesting spots pretty quickly without having a rental car.
The area of the Amalfi Drive deserves its reputation for beauty.
If you don't mind going an hour or so north of Rome before turning your attention south, I suggest Orvieto. Worth a full day or longer. If done as a day-trip, figure on the full day. If you want a fairly relaxing vacation, you could spend 2 or 3 nights in Orvieto.
It's difficult to estimate how much time someone else will want in Rome. I'd work from a guidebook, listing sights I definitely wanted to visit (not just walk past and glance at) and those on the maybe-if-I'm-nearby list. It's not just that really visiting a sight takes a lot more time--there's also the potential for time wasted in lines.
I'm more of a "wander and absorb the atmosphere" person and had already seen the time-consuming Vatican, so last year I spent just 4 days there, with a good part of one devoted to Ostia Antica (close, not requiring a full day). For a first-time visit, that would be too short for many people.
There are many other nice places in southern Italy, these are just off-the-top-of-my-head suggestions.