I've been to a fair few places in Greece, both mainland and islands - Athens, Thessaloniki, Halkidiki, Rhodes, Symi, Kastellorizo, Aegina, Poros, Skiathos, Skopelos, Alonissos, Lefkas, Meganisi, Corfu, Kefalonia... I agree that everywhere will be hot in August, however, the Dodecanese are very arid. I would look to the lusher Ionian islands. Corfu has something for everyone, Paxos and Anti-Paxos are stunning, Kefalonia is fab and Meganisi is teeny-tiny and very low-key. You can easily hop over to Lefkas from Meganisi if you wish to explore further.
The Sporades (Skopelos, Skiathos, Alonissos) have okay beaches, great food and there are boat trips to see the protected seals. Not the prettiest islands, although Skopelos Town is lovely.
Symi in the Dodecanese is amazingly pretty, but might be hard work for little legs as it is incredibly hilly/steep.
The last time I was in Greece was when I was pg six years ago. We stayed at Ekies Manor in Halkidiki and it was really lovely - I'd google it and have a look.