Route One
Take a walk on the many beaches found to  the west of the island which are all joined to each other
 Saint Prokopios - Saint Anna - Plaka - Orkos - Mikri Vigla
 We start at the beach of Saint Prokopios with its crystal clear water and white granulous sand extending for almost 2km.   Standing on a  hill the small church of  Saint  Prokopios dominates one end of the beachand it is here that you can enjoy the festival held every year on the 8th July
Continuing onwards we then reach Saint Anna beach with its fine sand and cedar trees which  give shade to  the sunbathers.  The local church  is called
Agia Anna and is sited near to the small harbour found in this village.  Here the festival is held on 25th July
Past Agia Anna the asphalt road  gives way to a unmade earthy road which is negotiable by cars and bikes leading on to Plaka beach and then onwards to  the further beaches of Orkos,  Kastraki, Pyrgali - all fine sand beaches with cedar trees offering shade -  and finally to  Mikri Vigla beach which was at one time the look out point for invaders or pirates arriving by boat.
2nd Route
Take a drive from Chora to Apeiranthos by one of two routes
Chora - Tripodes - Vivlos. Sagkri - Chalki - Filoti - Apeiranthos 
Chora - Galanado - Sagkri - Chalki - Filoti - Apeiranthos
From  Chora  take the road to Tripodes which is also known as Vivlos.  Here you will find three imposing windmills looking far out over the meadows and down to the Agean sea
If however you decide to travel the second route - Chora to Galanado you will pass  the Tower of Mpelonia standing next to the small church of St Ioannis takign in both Catholic and Orthodox elements. 
Continue onwards to Sagkri and here visit the Monastery of Saint Eleftherios and Saint Ioannis of Gyroyla and also the Temple of Dimitra.
Arriving in Filoti, the largest village of Naxos,  traditional  cafes and tavernas  and a church dedicated to the Virgin Mary of Filotisa surround  the large Plane tree - the meeting place for the villagers.
Leaving Filoti behind  continue to  the most northern tourist area of the island - here you will find the Cave of Za and the picturesque village of Apeiranthos  which is said to have been built by  the Cretans and the local dialect is very similar to an area near Rethymnon in Crete.   Try the traditional Patoydo  and explore the picturesque back streets in the village of Glezos and take time to visit the local Folklore and Natural History Museum and also the Tower of Zeggoli.
After all the sightseeing take a refreshing swim at the nearest beaches - Moytsoyna and Psili Amnos.  Here you can also find good fish tavernas.