Ballet / concert/ Broadway..
As a theater nerd, I'd have to say any show is going to be great! Really, truly, you can't go wrong. It sort of depends on what you want to see, though.
If you're not looking to see a particularly holiday-themed show, you see what tickets the TKTS Times Square booth (or TKTS South Street Seaport or TKTS Brooklyn) has. Depending on the day/time you could get really amazing seats for a show that you might not normally have considered seeing.
The Christmas Spectacular at Radio City Music Hall is great, especially if you're traveling with kids.
Check and see what the Lincoln Center might have playing. I saw an opera there once and while I didn't understand a word that was said (or sung, I suppose), it was fantastic.
And if you're not traveling with children and want to see something really unique, check out Sleep No More at the The McKittrick Hotel. It's Macbeth in a large warehouse where you're permitted to walk around by yourself. It's hard to explain, but it's such a cool experience. Very memorable!
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If you're looking for cheap, in addition to the TKTS lines, you can also sign up for the Will Call club at Studentrush.org (you don't have to be a student) and sometimes you can get rush tickets for a mere handling charge. Also, many of the top shows have lotteries for cheap tickets or SRO a few hours before the show. for example, Book of Mormon.
When I lived in New York, I made it a tradition to go see "The Nutcracker" at the New York City Ballet! They put it on every holiday season and it truly is magical. If you're looking for something a little bit more glitzy, I would go to Radio City Music Hall's "Christmas Spectacular" show, with the famous NYC rockettes!
Broadway is also always a good option and there are just so many shows to choose from. If you're looking for a more unique experience, I would defintely also check out shows off-broadway like "Stomp", "Fuerza Bruta", and " Blue man group".
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Claire - SRO is Standing Room Only
I understand Elton John is playing at Barclay Center on New Years Eve. Probably a tough -but fantastic ticket I think. As far as theater, many new shows beginning previews in October so it might be good to wait a month for some reviews then buy them. The best shows sell out for the holidays and resale prices very expensive.
I absolutely love lion king. I've seen it more than twice. You don't have to see it during the holidays but it is a great show!
The Cloisters has wonderful seasonal concerts: http://www.metmuseum.org/events/programs/events-at-the-cloisters/concerts-and-performances-at-the-cloisters
Really enjoyed the Rockettes Christmas Spectacular Radio City Music Hall last year. Look online for discount tickets