hey man, i live in bozeman but one of my best friends is from billings so he gives me the low down on the city all the time.
we all like to joke with him that billings is a shithole, and unfortunately, that's partly accurate. i've been to billings once and it is kinda a shithole-- right next to a bunch of industrial factories, etc. pretty ugly if you ask me. i don't think it really has a outdoorsy or ski town feel to it, which is unfortunate.
now that the shitty stuff is out of the way lets get to the good stuff. downtown is nice. there's some fun stuff to do there, but that's about it for the city i have heard. the time i spent there was for an edward sharpe concert and the people there did not know how to have fun at a concert at all and were pretty asshole-ish. good news is you aren't far from the beartooths, which are the highest range in MT I believe. It is definitely worth getting a season pass to Red Lodge, because from what I gather they have some good terrain and can get hit hard with snow. on the other hand, people call it "rock dodge" for a reason.
also, bozeman isn't far (about an hour and a half). so that means that it's give or take an hour and a half to bridger bowl and about two and a half hours from big sky/moonlight-- on a good day.
lastly, if you climb, the rims are the sandstone cliffs surrounding billings and they supposedly have some pretty good cracks there.
but yea, thats about all i've gotten from my friend who was born and raised there. i'm sure you won't have a terrible time and everything will work out fine, it just won't be as good as living in bozo ;) if you ever come to bridger or big sky let me know!