When in Chora (the town of Naxos), you can easily pick up a little book on the walks of Naxos. You can then follow trails after you take the bus or taxi to get to your starting point of your walk.
Some of the walks are very remote and we seldom met anyone. These walks are glorious and you come across beautiful scenery and Byzantine churches and monasteries. Also a great pottery where you can watch the pots and things being made. Little villages where there is all sorts of things to be seen and food to taste in the taverns. Proper walking gear is advisable and it is nice to go with some friends as some of the terain is very remote.
As you drive from Naxos Town to apiranthos (there's only one road, so you can't get lost, you'll see a sign for "Pottery Workshop". It's owned by Manolis, who has been doing pottery his whole life. He actually comes from a long line of potters. His whole family has been potters. It's a pretty amazing place. He doesn't speak Englsih, so be prepared. he took us in, and within a matter of minutes, made an incrediable pot, right before our eyes. We bought several things, that were very reasonably priced. He'll also ship it for you, but that's a story for another time. Well worth the trip.
Mt. Zeus is the highest mountain in the Cyclades, 1001m high. It is a relatively easy hike, the ascend takes approx. 2 hours.
The starting point is at Filoti, which you easily reach by a bus from Hora. On the way you walk past the Cave of Zeus, which is one of many places claimed to be the birthplace of the god. Bring a good flashlight to be able to explore the cave.
The views from the top were great (although we really couldn´t enjoy them, see travelogue). And that feeling of beating the mountain, refreshing.
See travelogue for more details Ascending Mt. Zeus
For route information and other hikes in Naxos see "Walking the Greek Islands: Naxos and the small Cyclades" by Christian Ucke and Dieter Graf
I almost don't want to tell anyone about this, but that would be really selfish. I think this is off the beat and path. Mikri Vigla beach is south of the main town of Chora, and it is a great little beach with crystal clear waters, and an incredible taverna right there on the beach. It's a little tough to get to, you have to drive on a dirt road, but it's well worth it.
Just drive along the road from Naxos to Appolon. Half an hour drive with a scooter (I don't remember the name....) on the top of a hill you can see a small lagoon-like beach. Go there! No tourists, no facilities, but your own beach! Just drive down the dirtroad to the end, and your there.
Scooter is rented downtown for 6-7 EUR a day pluss petrol.
Hire a bike in Naxos (a couple of good places there) and head south along coast. Plenty of off-road paths through cane fields and along beaches. A few km south you can find a few farm tracks over hills. We took no maps as usual, and got lost as usual. But great fun. Sometimes unsure whether of not we were on some farmer's private tracks - we got interesting stares from the livestock! But eventually when you come across a real road you can see which way to head back to town. Great day's outing in the countryside - just us and the cows!
We found Naxos pleasantly less crowded than other islands. On another day we took a local bus from Naxos town into the hinterland east, hopped off and walked several km back to town. Definite need for maps there, as you're wandering through the mountains! (and little villages).
Seeing the full moon through the gate of Apollo's Temple was a captivating experience for me. I had to wait nearly an hour till the moon raised and appeared inside the gate, but it was worth the waiting!
Apirathos is a tradtional village about an hour or so from the main town of Chora. It's a tough drive in a car, and even tougher on a scooter or motorcycle, but well worth the effort.