If you like Art, the Metropolitan is FANTASTIC.
If you have been to one of the better Natural History Museums, like The Carnegie Museum in Pittsburgh or the Smithsonian in Washington, I think you will find the Natural History Museum disappointing, but if you've never been to any other, it's decent.
Saint Patrick's Cathedral is beautiful. That is midtown right off the subway. If you take a cab up to Harlem, the Cathedral of St. John the Divine is awe-inspiring. [In fact, depending on how much walking you intend to do in Central Park -- which is huge -- St. John's is right off the end of the upper west side of the Park.]
The subway goes right down to the The World Trade Center/Memorial. Well worth seeing.
The subway also goes to both the Citi Field and the New Yankee Stadium. [But don't hang around in the South Bronx after the game.]
One of my favorite things to do is to just walk down Park Avenue from Central Park, through the Helmsley Building and into Midtown. It's the heart of the City, like walking through a Grand Canyon made by Capitalism.
LOVE THIS!!
Loved your last sentence.....going to do that this summer!!