We hired a car last time and found that to be the best way around. Sicily is massive and there's stuff all over the place so a car came in really handy.
We flew to Palermo and out of Catania but looking back I'd now pick the car up when we were ready to leave Palermo as driving in that place is mental. Looking back it's funny thinking about it but at the time I've never been as scared in a car. The rest of the trip was fine so don't let that put you off.
The history there is everywhere. The Greek ruins of Agrigento are stunning and definitely should be on anyones list when in Sicily. Went to Piazza Armerina, really cool ancient Roman mosaics. The Greek Amphitheatre in Taormina is also a great place to visit,.it looks down over the bay which adds to the beauty of it. Taormina is also a cool.place to.spend a few nights as well. Smallish town perched up in the hills. Obviously loads more and I'm betting others on here could go into more depth.
But 100% have lunch at Catania fish market