I grew up in Denver and practically lived on the mountains during college at CU. I think Copper Mountain is one of my faves for kids and it's where I grew up skiing. Breck is a great place to stay and close to Copper. The town is great to walk around and get food in. You could ski Breck too but I find for people new to that mountain it's can be a bit much. I also like Keystone but it can get super crowded so it's not the mountain I first go to with out of town guests. Beaver Creek is one of my other faves but it's pricey and a long drive unless you fly into some place closer. It's not as crowded as the other Vail resorts so that's nice and not as intense as Vail either. Steambeat is great but again a drive back to it. I skied a lot of more local type places..Arapahoe Basin being a fave while in college and we did ski Vail a ton but we had a student ski pass to a lot of mountains taking the sting away. My step-dad had a condo in Breck so I just love love that town. My mom lived in Aspen for a few years when I was younger so instead of babysitters while she worked she put us in ski school. (parents were divorced so we would go up for holidays and such) Crested Butte is another great resort but so so far removed.
I'd lean to Copper/Breck area or head farther in to Beaver Creek.
As for altitude sickness you should not get off a place in Denver and drive straight to the mountains. That's one of the main reasons people suffer from altitude sickness so bad. Stay a night in Denver and then drive up. It gives your body some time to adjust. Make sure you drink lots of water. Denver isn't a bad place to overnight in