Recommended villages:
Chalki - 15-20 mins from Chora - cultular and nice architecture, great to spend a couple of hours. There are some niche souvenir stores and galleries to have a look around AND some really nice food places. In Halki, i recommend ΣΠΙΤΙΚΟ ΓΑΛΑΚΤΟΜΠΟΥΡΕΚΟ which is a tavern with greek-naxian food and it actually has a great home made galaktompoureko and portokalopita. It’s preferable to visit before 3pm, as some stores are closed later on.
Filoti - 20-25 mins from Chora - There is nothing special about the village, apart from the main square where stands a HUGE platanos (plane tree) in the middle and the restaurants beneath it are nice. You can eat at Amvrosia, they have great new wave greek food. Reserve a table first.
Apeiranthos - 35-40 mins from Chora - Naxos’ most famous village, and it is well-deserved. Narrow streets to walk, stores and most of the places there are nice to eat or get coffee. I prefer Amorginos, a proper tavern with the menu being based on local meet (χασαποταβέρνα) or “Stou Lefteri”, quite place with greek food. Platanos is also nice, right in the middle of the village.
Suggestion for a full day trip
The villages above are actually in a row (the one after the other). Someone could start early and visit any of the above and then keep going after Apeiranthos to the East of the island where there are two really nice beaches, Psili Ammos and Panormos. If you find yourself there, you need to have beach-umbrellas with you, there are no sunbeds around. They are FAR AWAY (more than an hour) from the Chora, but it is a nice one-day trip around the island. After the beach, you can eat at Panormos (Mitato) or Moutsouna or any of the villages above, on your way back.
places i consider worthy for a drive outside of chora just for the food:
Vasilarakiou - Butcher shop in the form of Art. Vasilis has his own animals around the village, takes care of the whole proccess from…the livestock to bringing it to your plate and the output speaks for itself. It’s located to Kinidaros , around 25-30 mins driving from the Chora. The village is also famous for his local musicians (NIsiotika), so there is a chance to find an impromptu music even going on, although this could start later than midnight.
Vasilis (Melanes) - How local do you want it - Yes. Paidakia, patates tiganites, choriatiki, tzatziki, local wine, local cheese. His kids help with the service. His parents are in the kitchen. I think he doesn’t speak English. ~15 mins away from Chora.
Axiotissa - Naxos most famous tavern, it has been for years. There is a conflict in the chef’s mind between traditional and experimental greek cuisine but it works. Axiotisa can be combined with some of Naxos’ best beaches (see below for Alyko or Hawai). Important, you can’t just pop up in the place, you HAVE to make a reservation, as many days in advance as possible.