My friend and I stayed at a beautiful resort with a studio apartment at Irea Beach on Naxos. It wasn't too far from Naxos town and there is a good frequent bus service. This was the beach at Irea in front of our hotel, taken on a quiet morning before the tourists came to claim their beach chairs and umbrellas. It's a nice family beach and there are tavernas all along the shore where you can relax for dinner or cold drinks.
I think my favorite beach of all is Plaka Beach. The last time I was in Naxos, a few years ago, I camped at Maragas Beach, which is also excellent, right near Plaka Beach. From the studio at Irea you can walk here in about 15 minutes. I came early in the morning and there wasn't a soul. I recall my first visit to this beach (there were no umbrellas or beach chairs then) it seemed like I had been cast away on a tropical island. Fantastic! Even with the beds and umbrellas and a few more people than before it's still idyllic.
And there's nothing like an island sunset! I took this from a little taverna near Maragas Beach where I recall having a lovely sunset (solo) dinner on my last visit there. Even a solo traveler can't help but feel romantic in such a place!