This is the ideal way to explore Sicily. Modern highways circle and bisect the island, making all main cities easily reachable. The A20 autostrada connects Messina and Palermo; Messina and Catania are linked by the A18; running through the interior, from Catania to west of Cefalù, is the A19; threading west from Palermo, the A29 runs to Trapani and the airport, with a leg stretching down to Mazara del Vallo. The south side of the island is less well served, though stretches of the SS115 east of Agrigento are relatively fast and traffic-free.
You'll likely hear stories about the dangers of driving in Sicily. In the big cities—especially Palermo, Catania, and Messina—streets are a honking mess, with lane markings and stop signs taken as mere suggestions; you can avoid the chaos by driving through at off-peak times or on weekends. However, once outside the urban areas and resort towns, the highways and regional state roads are a driving enthusiast's dream—they're winding, sparsely populated, well maintained, and around most bends there's a striking new view. Obviously, don’t leave valuables in your car, and make sure baggage is stowed out of sight—in some cities it may be wise to keep the doors locked while in traffic. If leaving the car overnight, splurge on a garage.