When have skied at Copper and Winter Park I have seen a lot of little kids. In particular, there is a designated part of the mountain that is pretty much only easy green runs for kids to learn on. Sometimes these runs are even cordoned off so that the kids aren't in danger of getting run over by somebody flying by. Copper is even better in the sense that the whole west side of the mountain is only green runs and that makes it safe for kids.
However, both places have a really wide variety of skiing that extends back and beyond the part of the mountain you'd see from the place you're staying. Blues, blacks, trees, bowls, bumps, and if it's a good day of snow: powder. They got it all.
I have always liked the food choices I had and my kids would too, though as usual the prices at the mountain are expensive.
If you're staying slope side both have a lots of stores and restaurants, though lonewolf210 is correct that staying in Frisco and skiing at Copper (or whatever resort you decide is best for you) would make a lot of sense. Really, any of the big resorts in Summit County would be terrific. They all have lessons and activities for kids. When I went to Winter Park last year I actually stayed at YMCA of the Rockies - Snow Mountain Ranch. It's a little less fancy but I was quite impressed with the stuff they had for kids: big indoor pool, craft room for kids, sledding, ice skating and more.
btw: I grew up along Lake Michigan in South Haven and Holland.