CA is a good choice for skiing, I really like Mammoth and Tahoe skiing. For low cost and easy access I would suggest flying to Reno, staying there in a cheap hotel room and skiing Tahoe areas like Sugar Bowl, Squaw, Alpine Meadows or Sierra at Tahoe. I never have skied on the South Lake, but want to check out Kirkwood.
The beach and ski thing won't work unless there is snow near LA and that isn't a sure thing, more like a rare option. Mammoth is a 6 hour drive from LA if you drive fast, but the skiing is great. Mammoth is big, gets lots of snow, lots of snow stashes days after a storm, has a wide variety of terrain and there are some affordable places to stay like Bishop or Lone Pine.