Well now, here's a quick list. i'll add other things I can think of as and when.
• Staten Island Ferry - great way to see the Statue of Liberty from the water and the skyline of New York for free. YOu've got to get off at the end so make sure to move quickly so you don't get stranded on Staten Island.
• The Metropolitan Museum of Art - it's massive and you can get lost in there for days. I'd been there many, many times before I found the full size Egyptian temple in there. Think of it as the British Museum and the National Portrait Gallery rolled into one. Note that it's a 'suggested donation' to get in - so you can pay a dollar and get in. It's part of it's charter, so don't feel you need to pay their suggested donation of $20.
• Brooklyn Children's Museum - we don't have kids, but our friends who do, swear by this one.
• The High Line - beautiful raised park on an old elevated railroad in the heart of the city. It's close to the Intrepid, but it's also a good way to see some of the city with a little perspective.
• Right beside the High Line - Chelsea Market - think Borough Market with artisinal food places. Great for a coffee (or an amazing Jacques Torres hot chocolate). It's a good place to warm up before/after the High Line (though don't rely on it for bathrooms - the lines are almost always huge).
• Animazing Gallery - they have a wide rage of work on the walls, but if you hint you're interested in purchasing something, they have flat files full of original Disney plates, Tim Burton sketches and more. It's a treasure trove.
• Museum of Art and Design - they usually have fun architecture/design work and there's a great cafe at the top with Park views.
• Central Park (of course) - with specific destinations like Strawberry Fields.
Depending on where you're staying we can also provide restaurant recommendations. Just let me know - I think you've got my email?