It's been said before but I'll say it again: Go to Innsbruck.
Awesome college and party town with a large population of young student skiers and boarders from across the world in the heart of the Tirol region, which is some of the best skiing you'll find anywhere.
The Tirol Snow Card covers 86 resorts in the region on one pass and costs 696 euros (which is about 900 bucks).
If you want to travel, you can go west to Montafon, the Arlberg, and Switzerland; south to the Italian Alps; north to the German pre-Alps; and east to the Salzburg region. But you won't need to go anywhere outside the region if you don't want to.
You'll be able to find carpools easily, and fast public transportation to most nearby ski areas. And if you don't feel like traveling for 30 minutes to go ski, there's a ski resort on the Nordkette above Innsbruck with a lift that leaves directly from town. You can literally walk out your front door in ski boots and jump on a bus to the bottom lift station.
Switzerland is too expensive (unless you're rich), and France can be spendy as well compared to Austria.
Here are some other good study abroad destinations:
Munich, Germany (awesome big-city living, only an hour from the mountains)
Salzburg, Austria
Annecy, France