Catania down south-east has plenty of beaches and the best weather.... but there are nicer areas of the island IMO. We did a 2 week driving trip around the island and loved the whole place. Late-night nightlife outside of Palermo is hard to come by. So luckily for you, there won't be many boozed up punters anywhere in Siciliy, although everyone eats out and drinks in cafes late.
Cefalu up on the north coast is really nice, but I don't recall it having much of beach. Maybe ther are some nearby I didn't come across. But it has amazing restaurants etc..... And the town itself is beautiful.
Taormina is also fantastic, although it can be over run with tourists at times... But for old world grandeur (Cannes style) it's hard to beat. It's got a 1960s movie glitz about it. The beach isn't brilliant (quite stony) but was always busy.
Up around the N-E and near Trapani airport, the place is different again. Not as many tourists and the place is much more rural. Not a sinner speaks English in that part of Sicily, as far as I could gather. So if, like us, you have only basic Italian, brush up on it before you go. Erice, overlooking trapani, is nice for a vist. But again, I'm not sure of the beach count in that area.