I'll start by saying a week in Sicily is not a lot, so you'll have to choose what's most important. Also beaches are pretty but the water will be too cold for swimming (just in case you were thinking about it). I was there in late April 2 years ago and the wildflowers on the western side of the island were more plentiful and colorful that I've ever seen anywhere.
Rent a car in Palermo when you're ready to leave. With good driving instructions and some patience, you'll be out of the city and on the freeway in no time. Traffic in Palermo can be incredibly slow, but it's not so difficult that it's worth schlepping out to the airport to get a car.
On Sunday, take the bus to Monreale, don't miss the cloisters. On the way back, you can get off near the Palatine Chapel. At that point, I took the Frommer's self-guided Old Town Walking Tour (from their website).
On the way out, you could stop at the Zingara Reserve for a hike (4 hours), see the Greek temple at Segesta and get to Trapani or Erice for the night.
From there, to the south coast, stopping to see Greek temples at Selinunte, then Agrigento (excellent archaeology museum). If you stop at Selinunte, consider using the shuttle - the site is large and may save you time. If you don't stop at Selinunte, you could probably get to Piazza Armerina for the night.
Next day, visit the Villa Romana, then stop in Noto and/or Ragusa on the way to Siracusa for 1-2 nights (stay in Ortigia).
End your stay in Taormina. Skip Etna. Skip Catania. BTW it's easy to return the car at the airport. Be sure to fill the tank before you get there, and follow the signs carefully for rental car return.
