by boat from Piraeus, Rafina, Naxos, Syros, Paros and elsewhere
by plane on the route to Naxos, Paros, Mykonos, or Santorini and then hop on a ferry to Amorgos
There is a daily ferry direct ride from Athens heading to the two ports of Amorgos (Katapola and Aegiali), which is running throughout the year. You can either take the boat from Piraeus port (the trip with a conventional boat lasts about 9 hours/6 hours with high speed ferries during high season), or from the port of Rafina (11 hours/5 hours during summer)
The access to the two ports of Amorgos (Katapola and Aegiali) is possible by boat from Athens on a daily basis, all the way directly from the port of Piraeus (with a journey time of approximately 9 hours via conventional ships or with a speedboat in 6 hours during summer) and from the port of Rafina (about 11 hours, while during summer the trip lasts 5 hours via seajets).
Also, during summer there is a boat connection between Amorgos and the islands of Donousa, Naxos, Santorini, Syros, Paros, Ios, Kimolos, Anafi, Milos and Astypalea. Alternatively, you can get on an airplane from Naxos, Syros and Paros, and then hop on a boat to Amorgos.
All above info is prone to cancellations/delays all over the year.
For more details on timetables etc, contact:
Port Authority Station of Amorgos in Katapola: 0030 22850 71259
Amorgos Port Authority Station in Aegiali: 0030 22850 73620