I'm from NY, so I'll list a bunch of things that I feel are necessary for a good tourist experience here.
1. Visit Rockefeller Center. Take the tours, walk around, etc. And, whatever you do, go to the Top Of The Rock
(at night). It's one of the best views in the city and something every tourist should do.
2. Empire State Building. Look at it, go inside it, and whatever you do, visit the top of the empire state building!!! It's without a doubt, the best view of the city (and do it at night).
3. Visit all the little cultural "towns" and eat the best food there. You got Chinatown, Little Italy, Washington Heights (Spanish), Brighton Beach (Russian, there's an actual beach here that is quite nice with a chill boardwalk), and Harlem. Do some research and see what you'd like to see from each area.
4. Visit Central Park, and visit it many times and try to get through all of it. Each section of the park has its own little spark and flair and are noteworthy in its own way. Go to the Great Lawn, Sheep Meadow, eat at the Boathouse (an incredible restaurant with an amazingly scenic view of the lake), take your loved ones on a nice row trip on that same lake, go take a tour of Belvedere Castle. The options are endless in Central Park. They even have gondolas.
5. Go to Coney Island. Tons of things to see here, during the day and during the night.
6. If you like theater, you must invest in some broadway shows, maybe some off-broadway, whatever you'd like, New York has. If you like jazz, definitely visit some popular clubs. Off the top of my head, you got the Blue Note, The Vanguard, Iridium, Birdland (Go see Tommy Igoe!!), Smalls (Fantastic Place for be-bop!!!!!!), Pete's Candy Shop, Apollo Theater, etc. Also, be sure to take a stroll down Macdougal street, along the intersection at Bleecker Street to check out some awesome music clubs. Down there, you got Groove, The Bitter End and a ton of others.
7. If you like museums, definitely visit the Big 3: Metropolitan Museum of Art, The Museum of Modern Art, and the Museum of Natural History. Also, The Guggenheim is fantastic, and if you're a little kinky, The Museum of Sex downtown is quite nice.
8. Some lesser-known, but equally worthy attractions include....go to the Shake Shack. It's pretty much the best burger you can get in the city for the price of fast food. The most popular one is in Madison Square Park. It is on 23rd street and 5th avenue.
Go the Highline Park. It's an underrated, elevated park that can give you some amazing views of the Hudson River.
That's all I can think of at the moment. If I think of anything else, I'll revisit this thread. Sorry this was so long :(, but there's just an enormous amount of stuff to do here.
Btw, all this stuff is actually very easy to get to as long as you have an understanding of our (rather complex) subway system. Nothing is farther than an hour away, and, depending on where you're hotel is, most places are between 5 and 20 minutes away.