A week to 12 days is certainly do-able!!!
As far as what to see, if you're into archaeology, Agrigento on the Southern coast is a must-see, Syracusa has an excellent museum that I HIGHLY recommend, Piazza Armerina and Morgantina in the center are impressive (and fairly close to each other). On the east coast, Syracusa again has some fascinating sites and architecture along with the museum, Taormina, Mt. Etna (which was the one place we weren't able to visit, but drove by frequently). Near Taormina (to the south of the place) there is a museum and site at Giardini Naxos, as well as a beautiful harbor. On the north, Tindari is great, and you have a good view from their towards the Aeolie Islands. I don't know that much about the western coast and area, but others have suggested sites around there.
A nice day trip is to catch a ferry to Reggio Calabria on the mainland (if you haven't been there) and visit the museum there. It's worth the trip.
The Aeolie Islands (Lipari) is a lot of fun, there is an excellent museum there, and on the northwest coast there is a small site with a hot spring that used to be a bath. If you've ever had any experience with a motorcycle or scooter, rent a scooter and drive around the Island... it's worth it!!! Oh, and take the hydrofoil- it's fun and VERY smooth.
I was there last summer, and while we were busy during the week, we did all of this (and more) on the weekends (5), including driving from where we stayed (north of Taormina near Fiumedinisi) to these places.
Plan your stays and visits so as to go from one area to another, and you can get it all in- and if you drive, you'll see some fantastic scenery!