First off, if you're looking for a western US ski hill with a great daycare and kid's program, Brundage Mtn in McCall, Idaho is very hard to beat. We started our daughter on the snow 2 winters ago at Brundage at age 3. We rented gear for her for the season at a local ski shop ($50 for the season -- bring the gear back at end of season) and then spent a couple hrs of the first day with her on the bunny hill. Was a bit frustrating (for her and us), went to the ski school, asked around, and found the instructor that came highly recommended for little kids. Forked over enough cash for an hour private lesson (they won't take kids less than 4 in group lessons) and by the end of the hr she's skiing solo down the bunny hill w/o crashing AND LOVING IT. Spent another hr letting her show us her tricks, and she's had enough for the day. Remainder of the season, we'd make a point of skiing a couple hrs on the bunny hill with her (and at age 3, that's about all they have energy and attention span for) and then she's looking forward to the remainder of the day at the ski hill daycare with the kids. Got her a second private lesson near the end of the season, and before spring hit, she made a couple trips up the "big mountain" with us.
Last year (age 4, same Mtn) she's old enough for what Brundage calls their "Bear School". Combination of daycare and ski school. Check the kids in at 9, they'll play around for a while, hit the slopes with an instructor/babysitter from 9:30 - 11:00, back for lunch, slopes again from 1:00 - 2:30, and then hang out and play til Mom and Dad show up. Instructors/babysitters divide the kids into like skill groups, and everyone pretty much has a ball. We've managed our winter work schedules such that most of our ski days are mid-week, so when our daughter goes to Bear School, more often than not, she was one-on-one with the instructor, so it turns out to be equivalent to a 3-hr private lesson... Her skills and confidence level on the slopes far exceed anything I had in mind at this age, and she loves it. Best day of family skiing we had last year was at the end of the season after a spring storm. 16+ inches of fresh stuff, and we did a couple runs with her on a black diamond tree run. Nothing but laughter and fun for all of us.
KEY in this entire process though, is to NOT keep them on the slopes too much in any given day such that they get worn out or lose interest. If/when you start seeing the signs of tiredness showing up, don't con yourself into the "let's just sneak in one more run..." scenario. It will backfire badly, and you'll end the day on a bad note. Don't ask me how I know...
Can't say enough about Brundage Mtn -- Great family ski area. All of us are excited for this season to start. good luck!