A week isn't enough time even to attempt to see all of Sicily, so I think awrzesinski's suggestion is a good one. Stick fairly close to your landing spot, Palermo, rather than trying to get over to the east coast.
There's a lot to see in Palermo, so you definitely need to spend multiple nights there without a car. Driving in Palermo is to be avoided. But the car will be handy for getting to the cathedral at Monreale, just outside Palermo.
If you happen to be like me and have little interest in Greek ruins, you might include the medieval hilltop town of Erice outside Trapani. It's an extremely atmospheric place, though you should expect to have a lot of fellow visitors there.