We had one of our best holidays ever in Sicily - I'd go back in a heartbeat. First things first - it's MASSIVE. It's the largest island in the Med. I wouldn't entertain the idea of trying to 'do' the whole island. Choose a chunk of it and focus on that. We stayed east. We flew into Catania, then had a few days staying in Syracuse, to explore that area. Then went south and had a couple of days on the south coast - much more underdeveloped than many other parts of the island. There were great beaches there. From there we went to Modica - an amazing town - and had a couple of nights there, exploring all the towns and villages rebuilt after the earthquake in the 1600s. And then we had 5 days in Mascalucia - right on the edge of Etna - to see the volcano, and the surrounding area (and chill by the pool etc). It wasn't touristy at all - it was fab. We went on a day out up Etna in a 4x4 - hiking with a guide and going underground in lava channels.
The whole trip really was amazing - the architecture, nature, coast - everything. There are a couple of incredible geographical features - google Cava d'Ispica and Cavagrande del Cassibile. Both well worth seeing. There are endless lovely villages/towns to explore, and the food...
Most people seem to base themselves in Taormina if they're staying in the east, but I'd steer clear. It was nice to visit for the day (for the amphitheatre), but it's where all the cruise boats dock, and the package holidays go, so it's packed with tourists/day trippers and totally not representative of the rest of the island.
We stayed in a mix of SC and B&Bs, including a cave and a caravan in an olive grove, and an old monastery.