Don’t be thrown by the Metropolitan Museum of Art’s suggested admission price that is posted (unless they changed the sign) @jca is right you can give as low as $1 to get in. I usually give $5. I think the museum of modern art sometimes has a free day, you could check the website.
You gotta eat right? You can hit some of the famous places to eat. There are three delis: Katz’s, Stage, and Carnegie. Pastrami on the with mustard would be the most basic NY item to get. Dr. Browns cream soda is also like a staple in the delis. Very sweet soda, no caffeine.
Grand Central station is iconic, it’s used as a location in many movies, you can walk around yourself, or I think there is also a free walking tour available. Also, there is an annex for the subway museum inside Grand Central that I think is free. Little bit of transportation history and you can pick up cute subway trinkets.
Talking about the subway, stand on line at the info center in a subway station and you can ask for a free map. They usually have them, once in a while you might hit one that has run out of the maps so don’t get frustrated.
There is a skyway, or I think it’s called Highline Walkway? With gardens I think, and a grand view of the Hudson River. I haven’t been through that area in years, but I know it was being worked on several years ago.
Sometimes there are free concerts in Central Park in the summer time.
I think there is a free walking tour in Central Park.
You can take the subway for the price of a single fair (probably $2.50 one way) all the way out to Coney Island (the ocean) but I don’t know if it has been fixed up or is still very run down. I could find out, or a jelly might know.
If any of those interest you I’ll get details for you.
If you like “window shopping” Bloomingdale’s on 59th street is a beautiful store to walk through. Macy’s is famous for their windows along Broadway at 34th St.