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First, NYC generally requires a fair amount of walking so bring an umbrella. Hm, when I go outside, it's usually for shopping/eating. Don't really know your preferences but some general "New York only" places (and by New York I just mean Manhattan because I don't know the other boroughs that well...and by Manhattan my experience is limited from downtown to low-60s, mostly east of 6th Ave.) that you may consider:
South of Chinatown: Century 21 is a huge discount designer store that gets a ton of business, located across the street from Ground Zero.
Chinatown: Joe's Shanghai soup dumplings are really good. Chinatown Ice Cream Factory. Bubble tea places, following an above suggestion, that I know are TenRen, Vivi's, TeaRiffic, and Quickly's. (I like that last 2.)
Soho/Union Square/Herald Square/Times Square: shopping, being a tourist
East Village: Video Games New York (6th and 3rd)-small store but with basically every (console) game you can imagine, and the consoles too. And a cat. Better than a lot of other places I've been to.
Union Square area: Forbidden Planet (comics/anime/scifi/fantasy store). The Strand (bookstore). Max Brenners (chocolatier/restaurant).
Empire State Building/Grand Central station/Chrysler Building: just walk from 5th to Lex around 35th-40th and you'll see them all.
Bryant Park: Kinokuniya Bookstore (something like the largest Japanese bookstore in the mid-Atlantic region). Iconic branch of NY Public Library at 40th and 5th.
Upper East Side: Serendipity 3 (restaurant specializing in desserts, get the Frozen Hot Chocolate, was in the movie "Serendipity," may need a reservation at peak hours). It's across from "Dylan's Candy Bar," which people seem to like. (Never been because I always go to S3 at around 1am.) Chicken and rice cart (it's outside, but still worth it).
Broadway: you can see a show, but tickets are expensive. I still haven't seen one officially "on Broadway" yet.
Hope this helps and that you have a good time in the city.
South of Chinatown: Century 21 is a huge discount designer store that gets a ton of business, located across the street from Ground Zero.
Chinatown: Joe's Shanghai soup dumplings are really good. Chinatown Ice Cream Factory. Bubble tea places, following an above suggestion, that I know are TenRen, Vivi's, TeaRiffic, and Quickly's. (I like that last 2.)
Soho/Union Square/Herald Square/Times Square: shopping, being a tourist
East Village: Video Games New York (6th and 3rd)-small store but with basically every (console) game you can imagine, and the consoles too. And a cat. Better than a lot of other places I've been to.
Union Square area: Forbidden Planet (comics/anime/scifi/fantasy store). The Strand (bookstore). Max Brenners (chocolatier/restaurant).
Empire State Building/Grand Central station/Chrysler Building: just walk from 5th to Lex around 35th-40th and you'll see them all.
Bryant Park: Kinokuniya Bookstore (something like the largest Japanese bookstore in the mid-Atlantic region). Iconic branch of NY Public Library at 40th and 5th.
Upper East Side: Serendipity 3 (restaurant specializing in desserts, get the Frozen Hot Chocolate, was in the movie "Serendipity," may need a reservation at peak hours). It's across from "Dylan's Candy Bar," which people seem to like. (Never been because I always go to S3 at around 1am.) Chicken and rice cart (it's outside, but still worth it).
Broadway: you can see a show, but tickets are expensive. I still haven't seen one officially "on Broadway" yet.
Hope this helps and that you have a good time in the city.