sicily is a great choice for a vacation. as perche stated, it is not "touristy" and overrun. we have been a few times and love it
depending on where you are coming from (I assume the US), I would fly into Palermo and depart from Catania (or the reverse) to maximize travel time without doubling back
depending on how much "big city" experiences you want would affect your stay in Palermo. Frenetic but worth it. the traffic is horrendous. sightsee here on foot and then rent a car to leave the city
most of the sites (assuming that you don't want to move everyday (are in the south central and southeast of sicily. Piazza Amerina (roman mozaics), Agrigento (temples) Ragusa (old city) Noto (beautiful architecture) Siracusa (seaport) and Taormina (touristy but lovely - everyone goes there)