Bars: (if not too cold) rooftop bars such as Empire Hotel Rooftop, Hudson Terrace, The Press Lounge, Top of the Strand, offer great views
Music scene: I don’t know it very well, but I love jazz at Lincoln Center http://www.jazz.org/ Since DH and I are local we do the standby tickets which are super cheap at $10 released the Wednesday prior to a performance
Shows: Any Broadway show really! The hottest shows in town right now include The Book of Mormon, If/Then, Kinky Boots, and there are classics like Les Miz, Phantom of the Opera, Wicked, Mamma Mia, Lion King
Places for dinner (I’ll try to do these by prices):
Essentially you cannot go wrong with any of the 2- and 3-Michelin stars http://www.grubstreet.com/2014/09/new-york-city-michelin-stars-2015.html but they are pricy. My friends tend to prefer Eleven Madison Park, Jean Georges, Per Se, Daniel, and Momofuku Ko. Book wayyy in advance (call or use OpenTable.com or their online reservations), but if you mention to anyone in NYC that you dine there, it would be considered a 5-star, completely NYC experience.
Of the 1 Michelin stars, I’ve had great experiences so far at Ai Fiori, Bouley (their lunch tasting menu is awesome), Cafe China, Casa Enrique, and my friends are fans of Spotted Pig, Sushi of Gari, NoMad, Del Posto, Dovetail, Gramercy Tavern.
Otherwise, here are some other more NYC-type places that will not break the bank for you:
NYC pizza: John’s of Bleecker Street http://www.timeout.com/newyork/restaurants/johns-pizzeria-1
Dessert: Dominique Ansel for the Cronut (which I think is overrated), Lady M for their mille crepe cake, or Laduree for an overpriced French macaron
Chinese food: I looove the steamed soup dumplings at Joe’s Shanghai (multiple locations)
Burger: Shake Shack is kind of the burger place for NYC. Multiple locations.
Seafood: The Mermaid Inn (multiple locations) is great for oysters and seafood
Steakhouse: I personally don’t go to steakhouses just because they’re really really expensive and not as much variety of food, but, they are very NYC. Peter Luger Steakhouse and Del Frisco’s Steakhouse rank up there
Hope this helps!