Greece has more than a hundred inhabited islands and each has its own character, so choosing well is most of the work. Santorini is the famous caldera view and the busiest island. Paros and Antiparos are relaxed and easy with children. Milos is all boat days and strange white rock, and Crete is big enough for a week of its own.
We usually pair Athens with two islands at most, linked by ferry or a short flight. Mainland Greece is underrated: the Peloponnese has Mycenae, Epidaurus and the old town of Nafplio, and it stays quieter than the islands in high summer.
The honest caveats are August and the wind. August is the busiest month by far, and the meltemi, a strong northerly wind, blows across the Cyclades through July and August, making smaller boats uncomfortable and sometimes cancelling them. June and September are warmer in the sea and calmer on the water.