Hire bikes in Central Park on a Sunday (park closed to traffic on a Sunday)
Empire State Building
Guggenheim Museum (opposite Central Park)
Museum of Modern Art
FAO Schwarz - toy shop
Shops: Saks, Bloomingdales (department stores), Barneys (very expensive high fashion), lots of nice shops in SoHo, the Ralph Lauren shop is very impressive and there's other very smart shops nearby.
A wander round Greenwich Village (can be combined with SoHo)
Helicopter trip - helicopters take off from near Wall Street and Ground Zero. Highly recommend this as a honeymoon treat
Statten Island Ferry (then you don't need to go up the Statue of Liberty, it takes you right past)
Restaurants
Brooklyn - River Cafe restaurant looks back over to Manhattan (like the poster from the Woody Allen film)
Union Square Cafe - one of the top restaurants, not too expensive.
Mercer Kitchen in the Mercer Hotel in SoHo (lunch or dinner if you're in SoHo)
Gotham Bar & Grill
Café des Artistes: on the Upper West Side, not far from Central Park, it?s a lovely dining room
DB Restaurant Moderne is good for pre- or post-theater. One of the better ones in that neighborhood
The 21 Club is a little fogey-ish, but is such a famous restaurant.
Chinatown in New York is way different from the one in London and definitely worth a wonder through (stay away from Canal Street, which is just tourist tat). One of the better restaurants there is Oriental Garden.
If you?re up for a trip to Brooklyn, Peter Luger remains the best steakhouse in NYC
The restaurant at the MOMA (called The Modern) is very very good.
Gramercy Tavern and Union Square Café are two of the best.
There are so many restaurants to choose from in NY to suit all food types and budgets. For the ones above well worth booking in advance particularly for a Saturday night.
When you arrive buy a copy of Time Out New York magazine.- it will tell you everything that is on
Theatre
West Side Story: revival opening in a few weeks. It?s supposed to be fantastic.
South Pacific: also a revival, won loads of Tony Awards and has gotten rave rave rave reviews from everyone, even people who hate all that old Broadway kind of stuff
Guys and Dolls is also set to open in a few weeks.
I haven?t seen it, but a lot of people really like Jersey Boys. Not sure if it?s in the West End yet
Hope this helps. Have an amazing trip.