Well, if a kid has never been on skis before, downhill vs xc doesn't really matter - you just need to get them used to sliding. 6 is getting kind of old, but 4 is just about right. The Bill Koch Youth League starts at age 7, but most people who really enjoy skiing have been doing it since much younger.
Renting something like this
is a good start, because you don't need much equipment for them to start sliding around. No need for poles, in either downhill or xc.
The approach of a parent pushing/pulling the kid up a small (think sledding hill or smaller) hill, and letting them glide down, is about the level we're talking about.
As to downhill vs xc, once you get past the basic dad-pushing-you-around-on-skis level, that's more up to which sport the family wants to do. Good to have both skills, the fundamentals transfer easily, and as Janet says, the thrill of going downhill fast is a good one.
As for places... assuming the kids will quickly lose interest, you'll also want access to sledding and indoor stuff. Craftsbury is awesome, because of the lodging and food and snowshoeing and fatbiking and all the rest, but that's probably about as far from you as you can get. Staying somewhere in the Upper Valley near Lebanon gives you access to a wide variety of both xc and downhill, and is closer. Notchview xc in MA is pretty limited, but probably the closest thing for you to actually ski at. Not much else going on than xc skiing and snowshoeing there. Prospect Mt. in VT has really good skiing and reliable snow, and a nice lodge to warm up in. Dublin XC is free skiing, but no lodge and definitely no lessons. Windblown xc is near Dublin, and might do lessons, bonus that it has a [pretty wonky] orienteering map.
It's not a destination, but Weston is great for beginners and offers lessons for both kids and adults... and the Boston Children's museum and Aquarium and the Science Museum and and and...