I've been to a lot of the Cyclades islands. But it's been a long time.
Santorini is spectacular, scenically, but is now over-developed.
Same thing for Mykonos. Beautiful island, but terribly over-developed now.
A less-developed option is Siphnos. Paros and Naxos might be worth considering, too.
Outside of the Cyclades, there is Hydra and Poros in the Saronic Gulf. Not sure if there are any good beaches in Hydra—we just stayed in town. And we really liked Patmos in the Dodacanese islands—it's quiet and less developed, with some lovely beaches.
Crete might be an option—it has all the things you listed, but, again, I would avoid a resort full of tourists.
I've been to Greece three times ('73, '76, and '86) and traveled around it a lot. I love the place, and have been thinking recently of going back.
Try to avoid a package resort. Look for a hotel, or villa, or pension via the internet.