I didn't think Breckenridge was that great for non-skiers, personally, and the altitude was a real problem for me - I could hardly breathe the first day - so I certainly wouldn't take small DC there for their first experience of skiing.
This OP too is talking about two beginner DC and herself and DH as non-skiers - I just don't think it's very helpful to advise her to go all the way over the USA to try skiing. Her DC will spend the first 2-3 days on the nursery slope, which they could do absolutely anywhere, ditto another 2-3 days trundling down greens or easy blues.
OP I have another recommendation for you - Innsbruck or Salzburg. Both have direct flights from the UK, plenty of town stuff for you and DH to do, and nearby slopes for your DC to try skiing. It's very pretty there too - very chocolate-boxy Alpine views IYKWIM!
Don't assume that all secondary schools run ski trips, though. Or that if they do, your child will make it on to the list. My DCs school runs dozens of residential trips, but not skiing. As it happens, it doesn't matter for us, as we go every year as a family and I'm not sure how it would work out for my DCs, who are comfortable on black runs, to find a suitable ability group with the school. I'm sure they'd have a blast anyway, but at great cost.
There ARE skiing camps out there you can send your kids to (without you). Depends on how old they are, but the camps start quite young! Think of it as a 1-week boarding school with skiing being the only subject taught.