Ios and Amorgos are an appealing pair on paper: the lively beach island at the centre of the southern Cyclades and the long, rugged island to the east, loved by walkers and by anyone after a slower pace. On a map they look close. In practice the direct connection is thin, and knowing that before you plan will save you a lot of frustration.
In Ferryhopper's data for this direction there is a single company, Small Cyclades Lines, sailing once a week from May to September into Katapola, the main port of Amorgos. The trip takes around 2 hours 45 minutes to 3 hours, as the small ferry takes its time and may call at other islands, and the fare is low: from €11.20 for one adult when we checked. The high-speed Seajets boats that appear on this route only run the other way, from Amorgos to Ios.
That makes this route a matter of timing. If the weekly boat falls on a day that suits you, it is the cheapest and simplest way across. If it does not, the practical answer is a two-leg trip: Ios to Naxos and on to Amorgos, which works for much of the year, or Ios to Santorini and then Santorini to Amorgos in the warmer months.
Before you book, confirm the day of the weekly sailing for your month, check that your accommodation on Amorgos is near Katapola or that you have a way to reach it, and keep an eye on the forecast. A once-a-week boat leaves little room for error if the wind rises.