Thinking about taking the fam to see the Harry Potter play. We almost exclusively do outdoorsy vacations, so I need some help.
New York or Chicago?
What else should should we do? Museums, fun city stuff, etc.
What area should we stay in? Any specific places?
Kids are middle and high school age.
Thanksgiving week so I assume super busy and expensive?
Fallback plan is chilling in the desert 
plop yourself in midtown Manhattan around thanksgiving and you'd have a great time doing about anything you'd want. New York is honestly really special around Christmas and a lot of it is pretty cheap or free. Christmas markets, Macy's Windows, Rockefeller Center Tree, Saks light display, self-guided Elf location tour, all for free and all within a subway ride, most within walking distance from midtown/times square. Actually if you're there Thanksgiving Week you should go see them inflate the balloons before the Thanksgiving Day Parade around the Natural History Museum. (I will be doing that as well, also free). Ice skating in Central Park is great and cheaper/easier than Rockefeller Center, but I wouldn't blame anyone for wanting to do it at Rockefeller. Just maybe book that early and be flexible with times. There are also "free" walking tours that only ask for a tip that can be a good time. You should know that google maps does a great job with the subway and you can pay for any bus or subway ride with your phone or contactless card.
for museums I really like seeing how the richest and poorest lived by seeing both the Tenement Museum and JP Morgan library. But any of the better-known classics are great for kids middle school aged an older - Met, MOMA, Natural History, Whitney, Guggenheim, etc.
Broadway shows might be a little more expensive but if you spend some time looking you could probably find some silly cheap off-broadway christmas themed stuff. Some restaurants will be crazy but you can make reservations or always just get a slice of pizza or something from a halal cart if you don't care much about chasing food.
anyway I'd highly recommend NYC