My friends and I has been in the US for 3 months now, just skiing and bumming around like you will do, and I have some advise for you.
1.
Buy a car. If you're over for the season, you don't wanna be stuck in one place the entire time. And you can't really get anywhere without a car in the US, so having a car gives you the possibillity to move around a lot more. We've been to Breck, Keystone, Vail, Park City, and Bear (Colorado, Utah and Cali).
I am so happy we've been traveling around, you get to see a lot of the country, meet a lot of different people and I think you'll get way more out of the trip than if you just stay in one place all the time. Two friends of mine were in Mammoth for three months last year, and they really regret not moving around.
2.
Buy a camera, of course. They are really cheap here compared to Denmark (or Norway where I'm from). Make sure you get a PAL camera, not NTSC. check out http://www.bhphotovideo.com/ . they have everything.
3.
Check out Breck/Keystone while you're here. By far the best place we've been on our trip. We didn't go to Mammoth though, whick kinda sucks, but Breck is just crazy. Cool town too.
4.
Get fake ID's!! (I'm just assuming you're not 21 yet). search google for fakeid.co.uk..........
It has really helped us while we've been here. Believe me, it's soooo worth it. Just don't get caught, I don't think they take too kindly to that over here.
5.
Go to San Diego/LA or something like that and rent surfboards for a couple of days. None of us had ever surfed before, but it's definitely one of the coolest thing i've ever done. This is a MUST-THING-TO-DO while you're here.
6.
Get friendly with people. People here will talk to you whenever, and 99% are really nice people and loves europeans (especially the ladies). We've gotten to know a lot of people that way, which is a lot of fun. Chairlifts are a great place to meet people.
Hope my post was helpful, you should definitely do all the things I mentioned. And remember, no matter where you go you will have an amazing time. Enjoy it!!!