I only lived in Sicily for 1 year when I was 7, and despite my Venetian origins it somehow affected me, I am very fond of this place, it is probably one of the culturally densest and most varied destination I can think of, with better Greek ruins than in modern Hellas, and tons of Arab, Schwabian, Byzantine, medieval and Roman things to see, plus some incredible natural wonders...
While on the island, I recommend seeing the towns of Enna (off the beaten path), beautiful medieval Erice, the greek temple of Segesta, the amazing byzantine church at Monreale near Palermo, byzantine Cefalù, the incredible roman villa at Piazza Armerina, the old town of Syracuse (Ortigia), the high baroque towns of Noto, and Modica, the fishing town of Aci Trezza with its breathtaking sea views, and Taormina which, despite its jetsetty reputation is truly beautiful and remarkable, and of course go as far up mount Etna as you can it is stunning. Palermo ca nbe very interesting but it is a little rough, there are even inhabited neighbourhoods that have been left untouched since WW2 bombings, that in itself is kind of interesting...
The islands are among the most naturally beautiful of the entire Mediterranean, in particular Lipari, Salina, and incredible Stromboli, do take the night boat trip and the climb to see the eruptions!
Places to avoid: Panarea, an island paradise ruined by the obnoxious yacht crowd and turned into a shopping town for moneyed Milanese and Moscovite mafiosi.
The food is excellent, you will easily make it a habit to sip on an almond granita every morning, ice cream was invented here and you can find some excellent gelaterie. In Noto, visit Caffé Sicilia, paradise if you have a sweet tooth!