You'll have to pardon me, I have familiarity with SLC but I am not a local.
There is the Fireside Inn in Breck that is a hostel. Fireside Inn Bed & Breakfast and Hostel in Historic Breckenridge, Colorado: Welcome!
If you know your dates, you probably want to get a hold of them now about booking. I hear it gets booked up fast.
The Alpine Inn in Frisco seems to be a cheaper alternative too.
If don't mind driving a bit, Fairplay might be a good alternative for cheap accommodations. I think it is about 45 minutes to Breck from there. Warning though, the town is beat. Leadville aka Deadville is another spot and I think it is a little bit closer.
For lift tickets around Salt Lake. The premier spots are going to be the most expensive. Alta (no snowboarding anyway), Snowbird, Park City, etc. Snowbird is outstanding, Park City not nearly as good but they have a name. Don't overlook spots like Solitude, Brighton, or north of Salt Lake, Powder Mountain and Snobasin. Lift tickets at those resorts generally run up to around $60. Most of the ski shops in town sell discount tickets. Last time I bought a lit ticket for Snobasin, I got it for under $60. Both of those areas are sweet, and see much less traffic than the Salt Lake ski areas.
Salt Lake does have buses that run up the canyons. Sandy is the town to stay in from what I remember. There are several hotels with bus routes that go right by them. Again, the locals should be able to hook you up with this info better than I can.