short-cut to: New England, Pennsylvania, Colorado, Utah, Canada.
New York has two mountain ranges "upstate", with ski resorts, the Catskills and the Adirondacks; plus some smaller slopes, too.
Lake Placid, in the vast Adirondacks, was the site of the 1980 (and 1932) Winter Olympics; Whiteface Mountain ski resort is at the head of Lake Placid, with great views and highest vertical drop in the east, at 3430 feet. At Whiteface, Kids Kampus is a "one stop shop" with lift tix, rentals, kids programs; teens can do an all-day Whiteface Teen Experience.
Lake Placid also has an Olympic Center where you might see a Ski Jump competition, or ride up to the skydeck, or try some virtual reality ski-jumping yourself; or try the Olympic bobsled course. Or simply ride the town's giant old-fashioned toboggan run that shoots you onto Mirror Lake.
Further south in the giant Adirondack Park is year-round tourist area Lake George; families can ski at nearby smallish Gore Mountain, with 347 skiable acres in North Creek NY. Lake George has a Winter Carnival in February.
Closer still to NYC (just a hundred miles), the smaller Catskills Mountains has family-friendly Bellayre: small, with 171 skiable acres and vert of 1404 feet; operated by the state of New York on "forever wild" land. There's a Kidscamp, Nursery/Day Care, and a bus from NYC that includes your lift ticket.
Meanwhile, in western NY state there's Holiday Valley: "Consistently rated by visitors as the best ski resort in Western NY, Holiday Valley is a small, charming resort." Vert is 750 feet.
