I was couple of times in Greece, but still there are so many places (especially islands) that I would love to visit in the future.
I had opportunity to visit places like Athens, Thessaloniki, the Meteora and islands Santorini, Rodhes and Crete. I know that there are more beautiful island such as Zakyntos, Mykonos, but any of these and those not mentioned are really worth visiting.
Santorini island is said to be to romantic one. Mykonos is known as gay friendly and party destination, Zakynthos is well-known by Navagios beach (or Shipwreck beach), Corfu is great for families etc. Any way there are many suggestions and information you can get out there on the web – Telegraph made an article about The 19 best Greek island so you can check their opinion.
If you have two week for your holiday, maybe you will like this island hopping itinerary that includes 7 of the most beautiful Cyclades islands, suggested by Touropia – How to spend 2 Weeks Island Hopping in Greece.
I know everybody is concerned because of the immigration crisis in Greece but to be sure always check for the latest information, not every island had this problem.
For general information about Greece I can suggest website such as Visit Greece as they have information on where to go and what to see and do in Greece. As a Greek online island specialist check out the website Greeka they can also help you organize your trip.
The Guardian also has some articles about Greece that you can use as a source of information.
I can only say that beside the clear blue sea, in Greece you can enjoy great food, crazy parties, find adventures in the nature, relax on the sandy beach… just find a perfect destination for you.