Hi Claudia,
Yes, there are ferries from Kos Town to Kalymnos. You have two options, either the fast ferry (Dodekanisos Express) or the slower ferry (usually Blue Star DIAGORAS on the Rhodes – Piraeus line).
The fast ferry connects Kos Town to Kalymnos once every morning around 11. You can check timetables here: http://www.12ne.gr/en/
The slow ferry connects the two islands approximately 3 times per week. You can check for your specific dates here: http://www.openseas.gr/OPENSEAS/index_en.vm
Note that timetables are sometimes published only a few weeks ahead of time (!). So if you can’t find any dates now for April don’t panic, just keep checking (I know…not what you want to hear).
Mastichari is connected to Kalymnos port several times per day, but Mastichari is more than 30km away from Kos Town. There is a bus but I have no idea how regular it is or whether timetables are available online, and I don’t know how much a taxi would cost (sorry), but considering the distance it’s probably not very cheap.
So my opinion is to see whether you can get on a ferry from Kos Town, and if not, then catch the local boat from Mastichari to Kalymnos. This post on our website may be useful; it’s several months old but not much has changed: http://climbkalymnos.com/?p=3896
I hope this helps.
Katie