Amorgos and Ios are two very different islands. Amorgos is long and mountainous, the kind of place people come to for walking, swimming off rocks and quiet evenings in Chora. Ios is the social one: busy beaches, a hillside Chora full of bars in high summer, and good onward links south to Santorini and north to Naxos and Paros. Going from one to the other is a popular way to switch pace mid-trip.
In this direction there is a proper summer service. Seajets high-speed boats leave Katapola, the main port of Amorgos, on several days a week from May to September, building up to a daily sailing at the height of the season in last year's data, and take a little over two hours. Once a week, Small Cyclades Lines also makes the trip on a much slower, much cheaper small ferry.
The choice between them comes down to money and time. Seajets costs from €77.70 but gets you to Ios in 2 hours 5 minutes to 2 hours 15 minutes. The Small Cyclades Lines boat starts at €11.20, a fraction of that, but the trip takes about 5 hours 20 minutes and runs just once a week, so it only works if the day happens to suit you.
Outside May to September there is no direct boat at all, and the route home runs through Naxos instead. Whichever season you travel in, check that your ticket leaves from Katapola rather than Aegiali, the northern port of Amorgos, and allow for the drive if you are staying at the far end of the island.