The US is a big country (compare to Norway)
So, "visiting several places" can be intepreted in vastly different ways! That's not even counting Canada, which is even bigger!!!
On the other hand, resorts in the US are much smaller compare to those in the Alps.
So, you can stay in one place and ski MANY different resort within driving distance. That would give you a taste of what skiing in the US is like. But that doesn't give you a whole picture of what skiing in the US really is like. To fully sample the whole of N. America, you're talking about a lot of travelling and 3 weeks isn't even close to being enough.
Since nightlight and shopping isn't on your agenda, Salt Lake City would be the first anyone would suggest. It has access to about 10 different resorts within an hour of driving. Alternatively, you could stay in Park City and do the same, with a bit livelier night life. Either way, you're within driving distance (though not day trip) of Jackson Hole. This collection of resorts gives you the best powder and the most challenging terrain this side of the Atlantic!
The drawback is, none of the resorts are very big. You're missing out the mega-resort of Vail/Beaver Creek, and Whistler. You'll also miss out on Tahoe, where the views are priceless, the snowpack phenomanal and some terrain unbelievable!!!
Specific questions?