we have been all over the island and it is a fantastic location

assuming that you go in spring I don't think that beach days are in the cards. that being said, I would spend a few days in Palermo - it really is a fascinating city.

rent a car - it is the only way to get around, I would second the GPS idea. you can buy a chip here in the US to fit into yours if you have a unit

stop here for a selection of ceramics at good prices
http://en.wikipedia.org/wiki/Santo_Stefano_di_Camastra

head south to this. it is a must visit
http://en.wikipedia.org/wiki/Villa_Romana_del_Casale

here for spectacular greek ruins
http://en.wikipedia.org/wiki/Agrigento
eat here with a fantastic view of the temples. very good food
http://www.tripadvisor.com/Restauran...to_Sicily.html

beautiful town rebuilt after the earthquake
http://en.wikipedia.org/wiki/Noto

hillside town
http://en.wikipedia.org/wiki/Ragusa,_Sicily

siracuse is a great town with good food and sights

Catania is nice (somewhat gritty) but I wouldnt spend much time there when there is so much to see and do elsewhere. Palermo is a better choice for a big city. I would fly out of Catania though

the west coast of sicily is nice but more beachy