Avoid the Statue of Liberty. You will waste a large part of the day queueing. Instead take the free Staten Island Ferry. You get a pretty good view of the SoL from the water and also good views of the skyline.

An alternative is to take a helicopter trip. It’s expensive ($150-$200 for 15 mins) but IMO worth it. You’ll fly around the statue and also up the length of Manhattan.

Definitely take a walk through Central Park. Weekends are good because they close the interior loop road and you can skate or ride a bike around the whole park. While you are in that area, visit the Natural History Museum, and if you have time, the Metropolitan Museum of Art.

Visit the Empire State Building – but don’t bother going up (more looong queues). Got to the Rainbow Room instead, or the revolving bar/restaurant at the Marriott in Times Square.

Do visit Times Square. See a Broadway show if you can. Book in advance online if you can, or go to the TKTS booth in Times Square to get cancelled tickets on the day.

Experience a cab ride and the subway.

See a concert. Pick up a Village Voice on a Thu to get listings, or check here: http://www.musi-cal.com/search?key=city&value=New+York

Go to a Yankees game. But if they are playing Boston forget it – you won’t get tickets.

Do visit Greenwich Village and the Lower East Side.

Do eat a NY Bagel – especially from one of the famous stores, like H&H bagels.

More good advice here: http://www.concierge.com/ideas/hotspots/tours/500723?page=0