Walk the Brooklyn Bridge, go to Top of the Rock at Rockefeller Center. I love the parks —Washington Square Park and the High Line are my favorites and the High Line has great views of the city, and Riverside and Central Park are great. Not knowing your interests, it’s hard for me to recommend sights. I love walking the streets of Greenwich Village. Most of my attention has been outdoors. I’m not much of a shopper but I do love the Strand Bookstore. The New York Public Library on 5th Avenue is gorgeous. St. Patrick’s Cathedral is stunning but my favorite church in Manhattan is St. Francis of Assisi at 135 W. 35th Street.
Of course, the Empire State Building is iconic but for the best views I prefer Top of the Rock. Empire State is almost too high.
And, all the area at the 9/11 Memorial. At the very least, see the park area with the memorial pools.
I didn’t mention the Statue of Liberty because it takes a great deal of time and you can see it from land.