Don’t concern yourself with what’s touristy and what’s not. If it’s a cool/fun/great place, you’ll find tourists…who cares? People who suggest those two are mutally exclusive have way to much time on their hands.
Factors you should consider are when you’re going to be in NYC, how much money you have to spend and if you’re willing to travel to other boroughs.
With that said, some of the things I woud recomend and yes, some people might consider these things touristy, but it doesn’t mean they can’t be a lot of fun.
See a play or a musical. IMHO, you would best be served by seeing a fairly recent production rather than one that’s been around for a while. It’s not to say that older plays/musicals aren’t worthwhile just that newer productions have an energy that older productions lack. Also, don’t eat in any of the chain restaurants in Times Square, there are tons of great restaurants in area, go to one of them.
Go to the Metropolitan Opera. I know a lot of people don’t care for opera but to see it, in it’s full scale glory, is amazing.
Go to the Cafe Carlyle in the Carlyle Hotel. Bobby Short and Eartha Kitt might have passed away but there is still great cabaret to be had there.
Go to Bemelmans at the Carlyle Hotel for a drink. Ludwig Bemelmans, creator of the Madeline books, painted the murals on the wall.
Go to museums: The Metropolitan, The Museum of Natural History, Frick, MOMA, Cooper-Hewitt, Guggenheim, Whitney, New York Historical Society, Asia Society, PS 1, New York Historical Society and tons more (esp. the Cloisters as someone mentioned above. The Unicorn tapestries shouldn’t be missed).
See if BAM (Brooklyn Academy of Music) has any productions that might be of interest to you.
Go to Central Park and Prospect Park. Tons of things to see/do in both parks. Also the new Highline Park.
Go to the New York Botanical Garden (Bronx), the Brooklyn Botanical Garden, Queens Botanical Garden, Wave Hill and Staten Island’s Scholar Garden.
Go shopping or window shopping: Walk down 5th Avenue starting at 59th street and make your way to the New York Public Library on 42nd Street. The Library usually has some interesting exhibits for people to see. You can also go to any of the hotels you see along the way for tea or drinks (more than casual dress required). Behind the NYPL is Bryant Park, which has a great restaurant, kiosks if you just want to grab something to eat, tons of places to sit and people watch and other things to do. You’ll see St. Patrick’s too, the architecture is amazing.
Also there is the King Cole Bar in the St. Regis (again more than casual dress required) on this walk.
Just a note, I haven’t seen one in a while but occasionally if you walk this stretch (usually in the 40s), you’ll see a 3 card monte dealer. Never play. You will lose.
I like music/cabaret so it might not be something that interests you but the Duplex and Marie’s Crisis are two of my favorite bars.
For dive bars, just walk around the East Village and you’ll find plenty. The same for Alphabet City but I love 7B’s (on Seventh and B) also known as the Horseshoe Bar. Also in the East Village, you can try a place called PDT but you should make reservations first.
Walk around: East Village, West Village, Tribeca, SOHO (maybe get drinks at Pegu), Chelsea, the Meatpacking District and the Lower East Side. Tons of restaurants/lounges/bars/shopping.
Go to Harlem and Inwood.
Go to Morningside Heights and see St. John the Divine, the largest gothic cathedral in the world.
For eating. If you want to break the bank, there’s Per Se. I also like Babbo, Nobu, Blue Hill, Convivio, Alto, Bar Bouloud, Prune, Dovetail, Spotted Pig and again, many, many more.
The area around 28th and Lex. is called Curry Hill because there are tons of great Indian restaurants, among others, around here.
Go to Chinatown in Manhattan for good chinese food (also Bahn Mi a Korean sandwich specialty) or Chinatown in Flushing. Queens for even better Chinese food.
Steak houses aren’t my bag but they’re really popular for some reason. The best steak house is Peter Luger’s in Brooklyn. Anything else is 2nd best.
There are tons of places that lay claim to best hamburger but P.J Clarke’s rocks. Good drinks, as well and Timothy Hutton is there hanging out sometimes (he bought part of the bar a while ago).
The best thing about NYC is you can get practically any kind of food here, Ethiopian, Afghani, Ecuadorian etc. Try a cuisine you’ve never had before.
When you get to NYC, get a copy of Time Out New York. It’s very useful and will have tons of things for you to do ranging from free to very expensive.
Also, their website is a terrific resource, definitely look restaurants, bars etc. to check prices.