I'm hardly a western expert, having only been to two areas: Whitefish, MT (Big Mountain) and Salt Lake City (Deer Valley, Alta, and Solitude). Both areas are great options for skiing with kids.
I've been to Big Mountain several times, and have always had a fantastic time. The mountain is huge, with terrain for all ability levels, and they have a wonderful ski school. Positives were, really laid back feel, friendly locals, low altitude, uncrowded slopes, beautiful scenery (on a clear day you have an incomparable view of Glacier), neat local town (Whitefish) with free shuttle bus access, and total lack of pretension. Downsides were, can be really cold (our first visit was in mid-January, and it was -8 deg F and blowing 30mph on the summit), fog/low visibility, and more expensive to fly to since it's off of the beaten track. That last issue forced us to pick SLC for our most recent trip.
Salt Lake City is a great place to head with kids. It is a major airline hub, with easy access from almost anywhere in the US. You have at least 10 major ski areas within 90 minutes of the airport, and the 3 Park City resorts, 2 Little Cottonwood Canyon and 2 Big Cottonwood Canyon resorts are all within 45 minutes of downtown SLC. Check skiutah.com for more info. Stay down in town (Sandy) and rent a car to keep things on the cheap end of the scale. There is also plenty to do around SLC, if you want to take a day off from skiing.
My 10 year old son and I visited Dear Valley, Solitude and Alta. Deer Valley, while better than expected, was a little to posh and polished for my taste. Really expensive homes littered all over the slopes didn't give me any warm fuzzies. They do have plenty of terrain, buffed to perfection, and an extensive high speed lift system. Didn't get to experience the ski school, although they have an excellent reputation. We didn't catch Solitude on a great day, it didn't get below freezing the night before and the snow was very sticky. There were also some major flats, but on a better day, they would not have been such an issue.
We found our mecca in Alta. Having heard, for years, about their extensive ungroomed terrain, we were surprised by the amount of grooming on our first day. It was a little warm, and they groomed a bit more to compensate. And, despite their reputation, they have a great beginer/green run area. Unlike Snowbird, their next door neighbor, Alta can accomodate all ability levels from true beginer to clff jumping expert. My son had a great day in their ski school, and I received a killer, 2 1/2 hour afternoon clinic, almost a private, for a very reasonable $54. It was a great spring break trip, with an interesting ending (http://forums.epicski.com/showthread.php?t=40091)
We are looking forward to returning next winter. This time, the wife and 7 yo twin daughters will be in tow too.