Have not been but will probably check it out at some point  http://trop.ch/site/

you can get inflatable SUP's that go into a backpack as well

You can also hire them from Plage du reposoir. 

http://www.supgeneve.ch/p/infos.html  

I was wondering if anyone knew of a SUP paddle club that organises social group SUPs in the evenings or at the weekend?

check out eliquid on the net, they meet every wednesday afternoon in peney for some sup on the river, boards available for rent etc etc

I am a complete newbie at Stand Up Paddling but had visited Tropical (trop.ch) a few weeks back...not very expensive, around 20 bucks for an hour and good quality boards and paddles there.