There are several good ways to get to Kalymnos, most of them include a flight, a ferry and land transport combined. The quickest (and probably most reasonable) of them is to fly to Kos island, either with budget airlines or a charter flight. As of now (May 2019) there are several connections operated by Ryanair (www.ryanair.com ) including Berlin Tegel and Dusseldorf Int. (Germany), Rome Fiumicino and Bologna (Italy), or Vienna (Austria). Prices vary, but you can get a very good price if you’re lucky.
Another option is to fly to Athens – either with Ryanair (various connections available, but only from April 1st till October 30th), or with a charter flight. Once in Athens, take another flight from Athens to Kos (more frequent, more reliable and usually cheaper), or direct flight from Athens to Kalymnos. Although this is the quickest way to get here, it’s not the cheapest one (Athens-Kalymnos return ticket costs about €160) and flights are sometimes cancelled due to bad or windy weather (Kalymnos airport is small and the line is operated by small prop aircrafts). Both connections are operated by Olympic Air.
Once at Kos, you need to get to Mastichari port, which is about 7km north from the airport. You can get there by taxi (€12-16 per car) or by public bus (they go once every hour or so and cost €2 per person) Then you need to catch a ferry to Pothia (Kalymnos). Speedboat-like ferries cost €6 and it only takes 40 minutes to get to Kos from Mastichari, but waiting for a ferry can be a bit long sometimes (click for schedule, but keep in mind that it is not always observed) Last stage of the trip is to get from Pothia port in the east side to one of the villages in the west side of Kalymnos (about 9km distance). Again, you can choose between a taxi (€16-20 per car) or a small municipal bus (about €2 per person, it leaves from Pothia port every 2 hours. Buses operate from Monday to Saturday, no buses on Sundays)
If for some reason you can’t fly to Kos or Kalymnos, you can try flying to Rhodes islad , then take a ferry to Kos from there. In Rhodes you need to get to Mandraki port and catch a speedboat (3 hours sail) or regular ferry (5 hours) going from Rhodes to Piraeus with a stopover at Kalymnos (click for schedule). Once at Kalymnos, follow steps from section 1.
There are also several more exotic options to get here. You can, for example, take a ferry from Athens to Kalymons. In Athens, you need to get to Piraeus port, either by metro (Line 3 to Monastiraki Station, then Line 1 to Pireaus, total time about 1 hour, costs €6) or Express Bus X96 (direct conection to Pireaus train station, total time about 1h 15min, costs €3.20) Drawbacks: ferries are rare, operate only in Summer, and the journey to takes a painful 12 hours (schedule here).
You can also attempt to find a flight to Kos with another budget airlines, i.e. EasyJet (www.easyjet.com). There aren’t many connections available, and tickets are usually not that cheap, but if you are lucky enough, it may be a good idea. After that follow the steps from section 1 for the rest of the journey.