In reply to Daniel James:
1) Get your Ryanair Liverpool/LeedsBradford/Gatwick etc to Kos flight NOW before prices inflate.
2) Budget for taxi from airport to Mastichari (c15 euros) + ferry (6 euros) + Pothia/Masouri taxi (c18 euros). Double this for your return. Or save money using bus Pothia/Masouri (1.50 euros). Bus from airport to Mastichari will be cheap but no experience of this, sorry.
3) Another possibility if cheap flight to Athens is available is to then take magical Blue Star ferry from Piraeus to Kaly (c40 euros each way BUT airport to Piraeus is another c15 euros each way AND flight and ferries don't always match AND it's 12 hours on the ferry. However you don't pay for 2 Kos taxis or the Kos/Kaly ferry)
4) Apartments are around 20-35 euros per night for two. Plenty of apartments to choose from... easy to get in the Spring. Maybe hook up with someone beforehand and share cost.
5) Food: apartments have basic cooking facilities. Eating out is cheap, prob slightly cheaper than UK pub food.
6) On first trip, don't really need transport. There's enough crags within walking distance to keep you going for a long while. However, mountain bikes are about 4 euros a day.
7) Partners: post on UKC obviously but also on climbkalymnos.com (for more international companions). Also put notices at Kalymnos Climbing Centre and at Wilds Sports in Masouri.