Amorgos was my favourite place in the Cyclades and I really do recommend you spend a few days here when you visit Greece. The island is a bit more remote and less built on than the others in the group, which makes it a better place to come if you want to enjoy some peace and relaxation.
I rented a scooter so I could more easily and freely get around the island, but there are buses that run between the main sites of interest.
Amongst the things I most enjoyed doing on Amorgos were wandering around the windmills at Chora, snorkelling at the little bays you can reach from Katapola, visiting the wonderful Hozoviotissis Monastery, hiking in the mountains of Aigiali, bathing at the beautiful beach at Agia Pavlos, and looking out over the vast blue expanse of the sea at a restaurant in Katapola.