A few of my favorites (from someone who has been living in Manhattan now for almost 20 years, albeit the last 10 with young kids) (sorry for the length):
Central Park - just walking through it, stopping to listen to a random jazz quartet, watching a magician or puppeteer perform, taking a row boat out near the Boathouse and toward Bethesda Fountain, walking through the Central Park Zoo, climbing to the top of Belvedere Castle, walking through the Ramble and simply taking in the beautiful bridges and foliage, running around the reservoir, a horse carriage ride, skating in Wolman Rink (in the winter) with the skyline as your backdrop - a million things to do, all free or on the cheap, and never gets tiring
Brooklyn Bridge: one of the most picturesque in the world. Walking over it toward Brooklyn with downtown Manhattan behind you and then eating at Grimaldi's Pizza (if going around lunch) or Peter Lugers (if going around dinner) plus Brooklyn Ice Cream Factory as your reward at the end of the walk (there's also a beautiful carousel in Brooklyn Bridge Park that you should visit).
Either Top of the Rock, One World Trade or Empire State building for an amazing view of the city: They each have their pros. Top of the Rock has the best view and you can grab a drink at the Rainbow Room, Empire State Building has the history and cache and you can catch jazz (I think on Thursday through Saturday nights) (they also have a virtual ride inside), and One World is very modern and tech with the elevator with the virtual panels that show the City evolving as you go up, tablets that give you information on your surroundings as you walk around, etc.(and, of course, you have the 9/11 memorial right downstairs, which is a must).
Staten Island Ferry, Circle Line or the Beast: If you still want more photos of the skyline or Statute of Liberty (without actually going there - which I would forego), any of these are good options. Staten Island Ferry is free, Circle Line will give you a bit more of a tour, and the Beast (a speedboat) will essentially give you a nice tour and photo opp of downtown and the Statute of Liberty with a bit of a wet-thrill included. My kids ask to do the Beast every year and the tour guide has come to know my kids and knows that they love to be soaked with the super soaker. So, if you see a 10-year old in a PSU shirt, and you don't want to get wet - don't sit near him.
Museums: This is personal. Kids love the Natural History and the Hayden Planetarium, the Met is for anyone who loves art generally (a wide variety and huge), the MOMA if you love modern, and the Guggenheim if you like expressionist/contemporary and sometimes fairly eclectic exhibits or want to simply enjoy the architecture of the building.
Shows: Again personal. My current favorites out of the ones currently out would be Wicked (just love the theme and songs), Hamilton (cool juxtaposition of Broadway/rap music/American history), Lion King (great music and a spectacle), Book of Mormon (funny) and Aladdin. If you can get tix and are a Harry Potter fan, that just hit Broadway and gets great reviews (albeit it requires two shows). Haven't seen Bruce yet, but it's on my list and it also gets amazing reviews.
Food: Hard to go wrong here. If expense is not an issue, my favorites would be: Daniel, Momofuku Ko, Sushi of Gari, Il Mulino, Babbo, Dirty French (their bread is reason enough to go), Eleven Madison, Peter Luger. For mid-range, I'm a personal fan of pretty much any of the Tao places (Tao Downtown, Lavo, Beauty & Essex, Vandal, Bodega Negra, Rue 57).
