If considering Greek islands, Santorini is nice for a romantic trip but for families I would suggest Crete or Rhodes.
Crete has snow capped mountains as well as stunning beaches, and the palace of Knossos in Heraklion is fantastic. Seaside town of Chania is very pretty.
Rhodes has the old/ancient city inside the medieval city fortress, surrounded by the modern Rhodes town. My fave island. Beautiful beaches and lots to do.
Mykonos is nice when it is not high season - not crazy party town in low season (I agree that it is in high season), and very pretty. The ruins on the island of Delos (day trip from Mykonos) are amazing, but overall there was not a huge amount to do there in the low season.
On the Greek Mainland, the monasteries on top of the cliffs at Meteora are spectacular, and Delphi is well worth the visit too. Athens is a bit crazy, but if you like history is definitely worth a couple of days. I haven't been for 10 years though so can't talk about current conditions.
wherever you go, I hope your family has a wonderful time together. If I carry your bags, can I come too?