The easiest way to travel to Nisyros is to fly to the Kos International Airport (KGS), take a cab to the village of Kardamena (10 minutes, about €15), and take a boat to Nisyros (about 1 hour). There will be a boat on the evening of Sunday, July 13th (at 19:00, but may be adjusted according to flight delays). There are also daily tourist boats, departing at 9:30 every morning and returning to Kardamena at 16:30 in the afternoon, including weekends. Other boats itineraries from Kos and other islands to Nisyros appear in this spreadsheet. (Schedules are subject to change.)
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There are charter flights to Kos from all around Europe in the summer, and regular flights by Aegean Airlines from Athens (ATH) and Thessaloniki (SKG).​ For lovers of the sea, there is the alternative to travel to Nisyros by ferry from Pireaus,a 13h boat ride in the Aegean sea passing by many other islands. Book tickets and cabins here.​ If you choose to spend some time in Kos, here is a map of some interesting locations : https://goo.gl/6lvm0r.