Naxos is large enough that a rental car or scooter makes a real difference. The main road south from Hora reaches Agios Prokopios and Plaka quickly on good tarmac. The interior roads winding between villages are narrower and slower but perfectly driveable in a small car — some of the mountain passes are spectacular. The eastern coast is accessible but takes time: the road over the mountains to Apollonas and the north, or around to Moutsouna on the east, involves serious switchbacks and should be driven in daylight.
Rental agencies are concentrated around the port in Hora and at the beach resorts to the south. Book ahead in July and August; everything gets taken. Scooters are fine for the beach run and for a day in the villages, but not ideal for the longer mountain routes — gearing matters on those climbs.
Buses run from Hora to the main beaches and to the larger inland villages. KTEL bus schedules are posted at the bus stop beside the port. The beach services run frequently in summer (every 20-30 minutes to Agios Prokopios and Agia Anna), but the inland routes are sparser — typically two or three times a day to Filoti and Apiranthos, less to smaller villages. A bus can get you to Halki and back on a day trip, but you'll need to coordinate times carefully.
Taxis exist but are limited in number. The waterfront taxi rank has a few cars; for longer trips or early-morning arrivals at the port, it's worth having your accommodation arrange one in advance. The island is small enough that no trip takes more than 45 minutes.