Having traveled through most of Europe and all of Greece, I feel as though I have gained some great knowledge on this subject Naxos, the largest of the Cyclade islands is a wonderful destination to visit because it is very authentic and Greek, but not completely overrun with tourists. While most people will visit the famed islands of Santorini and Mykonos, Naxos is only about a 2 hour ferry ride a way and yet much less crowded!
Don't get me wrong, people from all over the world go to Naxos to enjoy the beautiful beaches, rich history and delicious food. The difference is that it's not overrun with college kids or backpackers like the other, above mentioned islands.
I've been to Naxos a multitude of times, and I've stayed all over the island. I recommend staying near one of the beaches, or staying right in the main town where the ferry leaves you off. There is also a small airport, if travelling by boat is out of the question.