It looks like I'm going to beat Prickly to this one, but I am paraphrasing his answer to a similar question a while back so he gets all the credit for the first bit.
1. Check out the Where to Ski and Snowboard website and make notes about all the resorts in Europe.
2. Make a short list of all the places you're seriously interested in.
3. Go to St. Anton.
There are plenty of fantastic resorts throughout the Alps, but with a trip to St. Anton you get it all. As SnowbirdDevotee mentioned, you can catch the train from Zurich airport, and you can get right to the center of St. Anton within a couple of hours. So, it's easy to get to and the journey is very scenic. Plus, there's loads of terrain to suit everyone, so you definitely won't be disappointed with that.
Now, some tips to keep costs down:
1. Go in the middle of January or early March. February is full of school holidays (similar to spring break in America) around the continent, which drives up the price of accomodation throughout the Alps for almost the entire month.
2. Fly out Friday (arriving in Zurich on Saturday morning) and return Saturday or Sunday. Most of the hotels and B&B's in town prefer or even limit their guests to Saturday-Saturday stays, so that will get you the lowest prices.
3. Use the St. Anton website for booking accommodation. It has the widest selection with loads of little B&B's and pension houses.
If you get a Sunday return flight, you can take the train back to Zurich on Saturday and spend a day sight-seeing there, which is a nice day out.