I was going to suggest Sicily as its generally cheaper than say Venice, but would also be hot in summer.
2 main airports- Palermo and on the other side of the island, Catania- so would depend where you could get flights to.
Cefalu is a beach area just over 1hr from Palermo. People rave about it, but I only read about it once we'd returned from 2 weeks in Sicily and didn't go there! From Palermo, you could also get a train to Agrigento which has ruins which looks similar to the Parthenon. The museum is interesting, but don't bother if you aren't history buffs.
From Catania you can get a bus to south Syracuse. We only went for the day, but there were some lovely restaurants and beach areas. North from Catania is Taormina which also has good reviews, but a bit more expensive. We didn't go there because Etna was rumbling so they said to avoid the area!
We don't have kids and didn't hire a car. Personally, I didn't want to risk of interpreting the parking signs wrong or not being able to get a park. We travelled via coaches or where available, the train. Both options were clean, on time, efficient and very reasonably priced. Both Palermo and Catania had the most amazing fresh food markets too.
If you need a family run BnB in Palermo, I recommend this place which was across the road from where the airport bus drops you off. The owner made freshly made cakes/biscuits and jams every day too. sicily.co.uk/places_to_stay/bb-extra-bed-breakfast/