I Island-hopped in my teens, and have also holidayed as a family - when I was the child - on the greek islands.
Koutoulafari in crete fab for families, small village, friendly people, no beach though. Crete is handy for daytrip to Santorini - scenic volcanic island. Would be hard work with young dc's though. My aunt favours the south of crete, but she travels without kids.
Ios is beautiful, but tailored towards tequila slammers/vomiting on streets set.(I loved it as a 20YO)
Rhodes caters for families and has lots to see, but the restaurants and pubs are desperate. Beaches crammed with tourists etc.
Naxos has plenty to explore but is hard work if you don't have some greek.
Have never been to the Ionian Islands, but plan on visiting Kefalonia(Captain Corelli obsession.) And Zakynthos(Zante) as it was recommended to me for family travel.
In general the Greeks are super friendly(too friendly if your a 20YO uni drop out.) Fascinating country, you cannot fail to be sucked in by the mythology and culture. Children are well received- as they are everywhere except Britain - food and drink in Greece not so cheap anymore. Weather fabulous in summer, crappy in winter.
Hope this Helps