You should plan a visit to Sicily around Palermo, Taormina, Siracusa, and Agrigento, four don't-miss destinations. The best way to see them all is to travel in a circle. Start your circuit in the northeast in Taormina, worth at least a night or two. If you have time, stay also in Catania, a lively, fascinating city that's often overlooked. From there, connect to the Catania–Ragusa toll-free autostrada (four-lane highway) and head toward the spectacular ancient Greek port of Siracusa, which merits at least two nights.
Next, backtrack north on the same highway, and take the A19 autostrada toward Palermo. Piazza Armerina's impressive mosaics and Enna, a sleepy mountaintop city, are worthwhile stops in the interior. Take the SS640 to the Greek temples of Agrigento. Stay here for a night before driving west along the coastal SS115, checking out Selinunte's ruins before reaching magical Erice, a good base for one night. You're now near some of Sicily's best beaches at San Vito Lo Capo.
Take the A19 to Palermo, the chaotic and wonderful capital city, to wrap up your Sicilian experience. Give yourself at least two days here—ideally, four or five. Or, if it's the warm season, consider heading east toward Messina and sailing to one of the lovely Aeolian Islands for a few days of leisure.