i'm biased cuz mama was born in Catania. The fish market there is a must and one of the best you'll ever see (and is much more than just a fish market). and of course Catania is right there under Etna (much of the city is built with black volcanic stone) which is also a must. If you have any interest in Opera, Catania is famous for Vincenzo Bellini one of the opera greats, and their Bellini Opera House (Sicily's most famous pasta dish, Pasta a la Norma, is named after a Bellini opera). and pay attention to all the elephant symbols in Catania.
Also, you should probably try the grilled horsemeat sandwiches available everywhere. Cavallo is tasty i promise. Wash it down with some great and unique Etna wines (it's own wine region).
I'll leave the rest of Sicily to others that know it better. One thing I definitely am planning on doing is seeking out the remnants of all the conquering civilizations that have run sicily in the past - muslim, spanish, roman. Sicily was basically Rome's resort island so there's plenty of that kind of history there.