As a beginner I would completley avoid Avoriaz, you will find the accomodation there a complete put off. There are many great places on the Swiss side of the mountains, I would highly recomend Champery for one.
The Crimbo week is expensive and crowed, however you can usually find great deals the week before (some good suggestion in the other posts). If you dont need to pre book and can decided a couple of days before, pick some where that has had the best snow cover. Champery is Chocholate box Switzerland, and has the advantage of the beginner slope being at the top of the mountain, which may sound a little scarey, but actually means that snow conditions are really good to learn on.
With the public transport the swiss resorts are much, much easier to get to and from, and you dont have to pay an arm and leg, again Champery is perfect as the railway station is at the main lift station and only a short walk to the bars, hotels B&B's - I can recomend fantastic cheap accomodation in Champery to boot. For the French Alps i use these guys
www.ski-lifts.com pretty good with the prices and they go to the main and not so main resorts and pick up from Geneve railway station if requested
For a first week I would look for somewhere with a good mix of night/evening entertainment and that the lift pass is not too expensive, there is no need to go to a big resort you simply wont get the value out of the lift pass.
Something that will help considerably with cost as a beginner;
A lot of resorts offer begginer lift passes that cover a fewer number of lifts, this is not often (if ever) advertised, and you will have to ask (they wont tell you) but means you get a considerably cheaper pass in most cases, and if you find you are natural's on the snow you can always upgrade the pass to cover a wider lift area after a day or two.
Athleticum - best place in Switzerland (based on price) to hire equipment, they do great begginer deals, and if you like the sport you can extend the hire for a whole season, usually for about the same price as a two weeks hire in a resort.
I have spent a number of winters guiding throught the Swiss Alps, feel free to contact me if you need more help or info getting organised