Take a walking tour of Chinatown and Little Italy. Chinatown has a million places for a cheap meal
Greenwich Village or the East Village are also nice to walk around. Lots of interesting people.
Central Park is also something you should check out.
I agree with all your suggestions, Sangha...and...
I just got back from NYC. We stayed for like 5 days.
For my suggestions...I would say:
Go to the docks...on the Hudson. Even take a 2 hour cruise from the Hudson all the way around to the East River...go up past the Brooklyn Bridge. Go to Liberty Island. Go to Battery park...watch the water taxis. Go to Washington Park...people watch. I like people watching. Some great park areas around Manhattan.
As you said, Sangha...Central Park...I love it. I can spend hours on end at different areas of the park. China Town is alright. There's a few "Dumpling places" that are fantastic. Can't remember the names right now. Chelsea area has some good places to eat.
They have these bleechers to sit on at Times Square if you just wanna sit and people watch. Go to MoMa (Museum of Modern Art) if you are an art fan. Don't go on Free Day...It's better to pay. You'd stand in line forever.
If you've never been...then catch Ellis Island, Statue of Liberty...it and Liberty are open again after Sandy hit.
Oh...you can go to where the World Trade Center buildings...were...and now that's a magnificent new building going up. And directly on the site is two memorial water falls with the names of all of the people who died there.
Go to the SOHO area...the artsy center of Manhattan.
Go to Motorino's Pizza...fantastic. Or the Meatball shop. There's great Deli's. But moderate priced places...quite a few that's pretty good.
I will tell you that there isn't really that cheap cheap food anywhere that's really worth eating. I found that really cheap cheap restaurants...you'll be disappointed with many. Better off at a fast food chain.
DON'T EAT around Time's Square...unless you want to spend a lot of money for food. Like say a sandwich, soup, and salad....55.00.
Some good food close to Harlem...and even the Bronx. Have to kind of check around on the net for some of those places.
If one's got some buck...that they don't mind letting go of...catch a Broadway or even Off Broadway play or musical. But it's an art to buying cheap tickets.
I love NYC...and I'm a country boy. I've lived in the country (Texas Hill Country) for 25 years. If I had to live in a big city...NYC...it'd be it...with San Francisco in 2nd place. Now that's opposite sides of the planet...but hey?