http://www.menupages.com/ is the best resource for finding dining in NYC that I've found. Pick a type of food and then the best-reviewed place. My wife and I used this site and ate at some amazing places. We saw Hitch when we were there on that particular trip and the place we were eating at was IN the movie - Tao, a Chinese Fusion place (the red place they did the speed dating in with the giant buddha).
No offense to tourism seekers, but the food is at least as high on the list as seeing landmarks. You can get the best of every type of food in NYC, and you'd do yourself a favor to really do some research before you go.
Other than that, here is what you should do and what I would do if I were going for the first time.
MOMA - the best modern art museum in the country in my opinion and a must-see. Has some of the most famous works in any US museum, and the museum ITSELF is a work of art.
The Met - huge museum, very diverse collection.
Natural History - If you're REALLY into natural history, you could go, but honestly, the one in DC is comparable if not better, and there's not much there that's really unique. The Met and MOMA are much bigger priorities if you ask me.
Empire State building - can take a very long time but the view is probably worth it. I'd do this over the statue of liberty to be honest.
Central Park - don't plan anything, just wander around (in the day time). Really quite beautiful and the pond with the little boats was a highlight for us.
NY Central Library - this is a really amazing library. Fun to wander around in for an hour or so.
Radio City Music Hall - really cool looking at night.
5th Avenue - FAO Schwarz, the Apple Cube, the shopping is some of the best there is and it's fun walking around there.
Times Square - go during the evening when it's dark. It's cool during the day, but it's amazing at night. You will feel like you are in the center of the world. If you see a Broadway near Times Square, take some extra time before and after to walk around and sightsee.
Broadway - I've seen Phantom, Wicked and Rent. Rent is a classic and is either no longer in production or will soon be gone, you should see it while it's there. Very New York and an awesome musical. Wicked is the best musical I've ever seen if you like more traditional musical fare, and Phantom was hardly worth seeing (for me). Spend the extra money and get good seats, you'll not regret it. You can't see anything of the calibre of Broadway anywhere BUT Broadway, and you must go see at least one, ideally more than one, while you're there.
Grand Central Station - Huge and busy, and the architecture is beautiful.
Leave a lot of time to just walk around. Go to the financial district, some of the famous churches in the area. Don't plan so much that you can't just wander around and soak it in. The great thing about cities like NYC and San Fran is that they are almost as much fun to just walk around than to do guided or scheduled activities. Very friendly for walking.