The Amorgos Bus Company (http://amorgosbuscompany.com) has timetables and ticket prices online. Summer buses go regularly from Katapola to Hora (€2) and Moni Hozoviotissis and Agia Anna Beach (€2), and less often to Aegiali (€3, 45 minutes). There are also buses from Aegiali up to Langada and Tholaria. Schedules are posted at the main stop in each village.
Without doubt, the easiest way to get around is with your own wheels. Cars, ATVs and scooters are available for hire from Thomas Rental or Evis Cars. Expect to pay from €40/30/20 per day for a car/ATV/scooter in August; less the rest of the year. There are only two petrol stations: one 1.5km inland from Katapola, the other in Aegiali.