I'll caveat this by admitting that I've never been to Utah, but I recently spent a lot of time researching the SLC areas as well of most of CO for a similar trip with young kids (5 and 3-yr old twins). Like you, kids ski school was a major consideration for me. With 3 kids to pay for, $$ was also a major consideration, which may or may not be the case for you. If it is and your budget won't cover Alta, you may want to check out Solitude. I personally ended up booking a trip to Copper b/c I had better flights into Denver, but my top pick in Utah, for my budget, was Solitude.
From what I understand, it's a fairly quiet area but, for a family trip, that may appeal to you and there are several relatively reasonably priced slope side condo options. There is also a shuttle you can use to get the little one to ski school and, if you rent a car, you could add a side trip or two in at other nearby resorts. Their kids ski school offers several mufti-day package options that are a good bit less expensive than most of the other resorts in that area that I looked at. Adult ski school prices seem like a good value as well. Again, I obviously can't speak from experience, but from what I've read/been told that their ski school is good to excellent.
The impression I get is that the main knock on Solitude is that advanced skiers may get bored there for an entire week. But as an intermediate myself, I'm not really sure that would be a problem.