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 €12), and take one of the tourist boats to Nisyros, departing at 9:30 every morning and returning to Kardamena at 16:30 in the afternoon, including weekends (€8 one way). The boat ride from Kardamena takes 1 hour. Other boats from Kos island to Nisyros include Panagia Spiliani and Blue Star ferries. Please note that Panagia Spiliani on some days of the week, and Blue Star every time, depart from the town of Kos, not Kardamena, which is 30km away from the airport and 2h away from Nisyros. Panagia Spiliani schedules are subject to change; usually, the final schedule for July is available by the beginning of June.
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 location : https://goo.gl/6lvm0r.