The Empire State Building is a must, but get there early (and if possible on a clear day!) to avoid the crowds. If you go in winter, wrap up warmly--it's very cold up there, especially with the wind-chill factor!
Statue of Liberty - The cheap way to do it is to take the free ferry to Staten Island (one every 15 minutes or so), which passes right by. You don't get to go inside, but the view is good, and you get a fab view of the downtown Manhattan skyline and Brooklyn Bridge.
Walk across the Brooklyn Bridge at sunset - beautiful.
At Ground Zero and the memorial in the park near it, there's not technically much to see, but it's definitely haunting when you stand there and think about it.
A show in a theatre bar (cheaper version of going to a Broadway show) is a great idea--the Duplex on Christopher Street is a place which houses such events.
Central Park, walking down Broadway and around Greenwich, and walking around the CBD at about 3am when no one is there, I felt perfectly safe, but it's really interesting to see a busy place so quiet!
The views from Brooklyn of Manhattan skyline
Chinatown--cheap food!
Quick Tips:
Definitely take in a show; however, if your budget doesn't stretch to that, see if you can find a cabaret bar, e.g. the Duplex on Christopher Street, which has live cabaret acts and a much smaller cover charge than a musical!
If you look/are young and are going out drinking, make sure you have your passport with you, as they ID at every bar.
Also, make sure you have a bladder the size of a few pints, as there aren't that many, if any, public loos - my travelling buddy had a very small one, and we made quite a few trips to various department stores--not to shop!
Best Way To Get Around:
We found that as we were there for a few days, the cheapest way to get around was to buy a week's travel card-- for 7 days unlimited travel--and use the subway and buses. The subway stays open all night, which is fabulous (especially as we're used to London tubes!), although I don't think I'd want to ride it on my own late at night as a girl, although I'm sure it would be okay. The map of it may seem daunting at first, but it's not really! If you want to go somewhere quickly, get an express train, which just stops at the white coloured stations. Also know if you are going downtown (towards the CBD) or uptown, toward Queens! The streets are eto west, and as the street numbers increase, you are travelling south to north. Very logical!