You should buy everything you're able to move in the States before coming over, but especially the clothing is probably good to buy overseas (depending on how fast your kids grow actually..).

I'm a snowboard instructor and I know that the city's ski and snowboarding school (http://ess-geneve.ch/) has a bus that always leaves from the village towards France - half an hour bus ride and then they are in the resort. Sounds like a great option for your kids (if they can still profit from lessons as well).
You can also take the train to for example Leysin (Great resort!) which takes 1h45.
Getting to the closer French resorts is probably easy with a ride too once you know people, and there are web pages for it too: blablacar.com.
Otherwise there's a bus leaving to Chamonix (Montblanc) a couple times a day for around 60 bucks with equipment back and forth I believe.

I don't know anything about the Jura mountainrange resorts, maybe someone else can tell you more about that.

And depending on where you will work there is also a bus every weekend that is organized by UN people (I think everyone can join though if you sign up - not sure, someone else could help here) that goes to really awesome resorts all over the place as day trips.

Hope this helps a little! :)

Dear Konrad,

This helps enourmously! Thank you! I am still looking (desperately) for a place where to stay but as soon as I solve that, I'll take care of the ski equipment for the kids. Yes, I am going to work in the UN area, so if anybody knows more about that bus trip - many thanks...

Thank you again!