You can't see all parts of Sicily in seven days (that requires more like 2 to 3 weeks), so you'll need to focus on what is most important to you. Pick up a guidebook to the island and dig in. Palermo has a lot of interesting sights (markets, churches/chapels, archaeological museum). The #1 sight may be the cathedral at Monreale, just outside the city (city bus service available). Some folks really like the former fishing village (now tourist town) of Cefalu, which is on the coast east of Palermo and easily reached by train.
Although Catania has a very attractive historic center and an interesting food market, as an east-coast destination it tends to take a back seat to Taormina (beautiful town in a magnificent setting, plus a Greek theatre, but with the tourist hordes to match) and Siracusa (lovely medieval district, a Greek theatre, an archaeological museum, and good side-trips to Baroque towns in the interior).
I'm not sure a car will be very helpful for getting from Palermo to Catania, because there is rail service (taking as little as 3 hours). There's also rail service along the east coast from Giardini-Naxos (just below Taormina) to Catania and on to Siracusa. Given the few days you have available, I'm not sure you're going to have time to do much outside the cities I've named, so the car may be mostly just something you have to pay to park.