Amorgos' most intersting site is perhaps the dazzling white Monastery of Chozoviotissa (alternate spelling: Moni Hozoviotissis) founded in the 11th century by the Byzantine Emperor Alexios Comnenus or in 800 AD by monks from Palestine's Hozovia depending on whose books you read.
You can either take a foot path from Hora to the monastery, or a bus. Even the bus leaves you with a 20 minutes walk. The walk is amply rewarded with superb views of the sea, so it's not a bad as you might think. It is a rubbly path, so wear good shoes. To enter the church, modest dress is required.
Once housing 100 monks in its 50 rooms, it now has only a handful. There are two churches and a library with 98 manuscripts. There is a small museum. The museum holds the miraculous icon found in the sea below the cliff, believed to have arrived from Asia Minor (or perhaps Cyprus or Jerusalem, again depending on whose book you read).
Below is the beach of Aghia Anna which is popular with naturalists and nudists. You can get to this beach on the bus.
Aegiali or Ormos (Bay) is Amorgos' northern port and its main resort due to the fact that it possesess the only sandy beach on the island. The town is small, relaxing and picturesque with striking views to the islet Nikouria. Nikouria is a granite lump of an islet, once a leper colony.
This picture-perfect little town is in a horseshoe bay and boasts a blue-domed church and isolated sand or shingle coves.
Over the headlands, paths lead to the group of isolated sand or shingle beached coves where nude bathing is optional. Caiques also are available to ferry you to quieter beaches. The Amorgos diving center offers diving lessons and equipment rental.
From Aegialia, take the bus to Tholaria, a pretty hill village with a vaulted Roman tomb, and to Langada, a pretty village under a rocky island thumb. From Langada, there's a path along the ridge that leads to Tholaria and the port. You'll see decapitated windmills, the frescoed church Yero Stavors, perhaps some donkeys in the valley, and a monastery that is a replica of Ag. Sophia in Istanbul.