Have been to Athens, Mykonos, Santorini. Santorini was our favorite. What islands are "must see" and why. Not beach people--history and archelogical interests--not to mention good food! Thanks!

1/ A tour(plenty of ferries and a flight connecting them) of the 3 north Egean islands Lesvos,Chios, Samos
Why? history,local culture and traditions,traditional architecture,and incredible local cuisineand bwines and spirits (total quality of local 'bio' fruits,veg.,delicatessen), and finally one of the best value for money in Greece,
A plus: that you can cross the sea strip and have a day tour in Turkey like Izmir,Cesme,Ephessus etc...

2/ a Tour of Peloponissos
Why? THE history! scenery combining beaches and bfabulous mountain villages, incredible food quality and high end luxurious resorts and private lodges.
A big plus: the fact that you just have to rent a car and no planes or ferries...

3/ Crete.
Why? Because Crete is Crete and is UNIQUE!Like a self contained nation.
cretan history,cretan food!!!!,beaches, mountains,and proud and smiling people.

I am in total agreement with the above and would add Karpathos because of the town of Olympia which still has active windmills, traditional ways, good restaurants etc.

I love Rhodes for a number of reasons. Rhodes City has the Byzantine architecture and a medieval castle. Also there's the old town which will make you feel like you've stepped back in time. Rhodes has a casino, if you are so inclined. Also you can take the ferry for a day trip to Marmaris, Turkey. There's another beautiful town on the island called Lindos. Lindos has all whitewashed houses and an acropolis with a fantastic view.

Agree that Rhodes is terrific. There are several small hotels to stay in the walled Medieval Town, which tends to be quiet at night after the cruise ships have departed. A trip to Lindos to see the Acropolis is fun. There are several other villages on the north coast past the airport that would make an interesting day trip.
Kos and Symi can be a daytrip from Rhodes using the Dodakannese Express Fast Ferry to visit.
Crete is a large, diverse island. Heraklion has the not to be missed Archeaological Museum and the Knossos site is about 15-20 minutes out of town on the bus. Chania is a lovely city in western Crete that you can use as a base for visiting the Samaria Gorge. Rhymenon is near some of the most beautiful beaches on the north side.
Corfu is entirely different with all the greenery and flowers from more frequent rains. Corfu Town is Venetian influenced (the Turks never ruled here). The food, architecture, atmosphere reflect this difference. Paleokastritsa is a beautiful beach northwest of Corfu Town. Also the Kanoni Penisula area that is about 3/4 hour walk from downtown has great views and you can visit the famous island on all the posters.
I've also been to Hydra and Aegina which are great day trips from Athens if you don't have time to go further afield.
If you have interests other than beaches and parties, I think the above islands have much more to offer than Mykonos and Santorini, but tourist officials seem to push M&S on the public. I can't figure out why adults go to Mykynos and Santorini doesn't have much interest except for the great views - really isn't alot to do there in comparison with the bigger islands.