I took a week in Sicily last year before making my way up to Milan, and it was terrific!

To sum up -- Sicily has all the spectacular sights and places of the north, without the crowds, with lower prices, and friendlier people too.

I would suggest traveling by train, unless you really want to drive somewhere in particular or have lots of kids to keep happy. In general, 2nd class tickets will be sufficiently comfortable for short journeys (and because in Sicily, the trains aren't state of the art anyway). For the route from Palermo to Messina or back, you may want to spluge for 1st class (air conditioning).

When I was there, visited Syracusa (impressive ruins, great scenery especially along the harbor front), Giardini/Taormina (Greek theater, nightlife in Giardini), Palermo, west from Palermo to the Egadi Islands, Levanzo (island, really charming, small town, beautiful water, warm dry breezes from Africa), Erice (mountaintop village with amazing views, quiet streets, castle), and then went north to the rest of Italy.

After visiting Sicily, I found the Amalfi coast a waste of time (kind of), with too many tourists, inflated prices, and the same things to see. I didn't really venture to the interior of Sicily, but I might have if I had had more time.

Where are you flying into? I'll try to post more here for you if I start to remember more details (prices, specific recommendations, etc.)