Cham / Verbier / Avoriaz / Morzine all quick from Geneva. Private transfer companies available, I have got the train to Verb before also. Sometimes cheaper to fly BA once you factor in baggage & ski costs etc on sleazy jet.
From Turin Cervinia is a great option, with lovely Italian food. BA and I think Ryan Air fly there, beware hidden costs on the latter though, they charge a lot for skis / boards.
The disadvantage of going for a long weekend is that you end up spending almost as much money as you would if you did a full week. If you're short of time then it may be worth it, but I'd rather go for a week if it was me!
I did a lot of weekend trips with DH before we had kids. We always travelled independently - Easyjet, taxi transfer, hotel, ski hire, lift pass, etc. There are a few companies that now do ski weekends - SkiWorld used to, but I think there's a lot more choice now. Do an online search.
Last time, flew Easyjet to Geneva, hired a car and booked an apartment via ownersabroad in Chatel.
But you can get to lots of resorts easily from Geneva - eg Chamonix is very close, and Morzine seemed good - someone staying there recommended an airport transfer service which means you wouldn't need to hire a car.
We have previously driven down for a long weekend, stayed in Brides-les-Bain at the hotel Altis val Verde. A 25 minute gondola trip then brings you out in Meribel.
I have to choose - a week with the kids or a long weekend with the girls. Have been with kids a couple of times and so I know what to look for but not sure where to start when thinking about a long weekend. Any tips anyone?