We did this at Easter couple of years ago - flew into LA, had a couple of days there with friends who live there, then drove up to Mammoth, stayed in a self catering condo right next to the slopes, ski in ski out (with hot tub and pool, which you skied past to get to the locker room).
Unusually for Mammoth it was not, by their standards, a great snow year, but it dumped while we were there, so we were skiing knee deep powder on piste. Great resort, not vast by European standards but plenty of skiing for us, only thing I hated were the chairlifts without bars (or the ones with bars, but the Americans don't use them!). Food mixed, some great, some v typical American, but lots of choice of places to eat. Free ski bus within the resort, we were staying at one end, Eagle Lodge, but we had a car anyway. I bought lift passes and ordered skis etc through a company here and got a good discount I seem to remember.
Our children were 15 and 13 at the time, so a bit older than yours, but loved the private lesson they had with an instructor. We had two weeks in total, so after 5 days at Mammoth drove down to Death Valley for a couple of days in the desert, before heading back to LA to fly home. It was one of our best family holidays ever! Obviously if you go west coast USA there is more jet lag to factor in.