From Geneva you can go either to France (Portes du Soileil, Tignes, Les Trois Vallee, Chamonix, Les Arcs, La Plagne, most have them have Shuttles from Genenva) or in Switzerland to Verbier, Portes du Soleil, or the smaller resorts of Leysin, Gryon, Les Diablerets. All of them are more than 1 hour from Genenva, rather 2-4 hrs. The Swiss resorts you can all reach by train.
Forget about cheap lodging, will be really hard to find in Switzerland. Cheapest is to book a whole week hotel + ski pass in a French resort.
In December snow can be sparse in this part of the Alps.