Try really hard to make it 4 weeks rather than 3. You could spend 3 weeks in Sicily alone even with your own car.
I haven't driven at all in Italy but have traveled around it quite a bit. I think you would enjoy Puglia if you have time to divert from the direct path between Rome and Sicily. There are lots of interesting small towns to explore. With a car you'd probably be able to wander around three of them in a single day. And the city of Lecce is full of Baroque architecture; it's worth more than a day.
On your way west toward Sicily you can stop in Matera to see the cave houses.
Remember that Sicily is somewhat awkward to reach, so try to give it its due. It would be a shame to go all that way for just a quick visit. My top tip there is to take advantage of having the car and see at least some of the main inland sights. They can be awkward to reach by public transportation, but you won't have that problem.
Ortygia, the historic part of coastal Siracusa, is a must. It's not a long drive south from Catania, but I'm not sure about taking your car into the old city. Using Siracusa as a base, you can visit Noto, Ragusa, Modica, and others that may sound enticing in your guidebook. The Villa Romana del Casale and Greek ruins near Morgantina are both accessible from Piazza Armerina, an inland city probably reachable from either Catania or Siracusa. I enjoyed Enna, too. It's an inland city near the middle of the island.
I hope you'll have time for Palermo. It's full of incredible Baroque churches and other religious buildings. It's a bit like Naples on steroids. The traffic isn't something I'd want to experience behind the wheel, but you might be OK with it. I'd say some advance research and planning on that point would be very worthwhile. If you do get there, don't miss Monreale, which is just outside Palermo.
Catania's a large city, too, but the traffic seemed a lot tamer than in Palermo. I found Taormina, which is north of Catania, touristy enough to affect my enjoyment of the city, but it is undoubtedly beautiful. You shouldn't go to Catania without seeing Taormina. Check into the driving/parking situation around Taormina. The core of the old city is pedestrian only and there are streets that are actually steps. It might be better to leave your car down near the coast and take the bus up.
Geography reduces your chances of seeing Trapani, because it's all the way on the west coast. But it's a very nice small city with a funicular providing access to the stunning hilltop Erice. I liked this combo much more than Cefalu (east of Palermo and quicker to reach), but Cefalu is also worth a visit, or an overnight.
The food in Sicily is fabulous. It's quite different from what you get on the mainland. Stronger flavors. And the gelato and granite are to die for. Noto is the classic place to try the breakfast combo of granite + brioche.
