Someone asks for something to do outside of Manhattan on their third trip here.
What time of year are you coming? You could go down to City Hall and walk acros the Brooklyn Bridge. Once there, you can exit the staircase where the Bridge flattens out, and head down Fulton Street to the Fulton Ferry dock. Along the way you'll see an excellent pizzeria by the name of Grimaldi's on the right side, and an old, large orange-brick buiding with an arched brick opening on the left. That building was the Brooklyn Eagle Newspaper Building, and they once had an editor named Walt Whitman.
Down at the dock, the rail around the perimeter is emblazoned with the words Whitman wrote about New York, "O city of spires and masts! My city!"
Stand at the dock and ask someone (others are there) to take your picture with the skyscrapers of Lower Manhattan, or the Brooklyn Bridge, or both as your backdrop.
Just a few steps away is The River Cafe, a restaurant on a barge. I'm not saying it's expensive but my wife and I went with another couple there for coffee and desserts, and the bill was $74. Bring lots of money.
If it's high summer, you could take the subway to the beach. Take the Q or B to Coney Island, a bay beach with small waves and brackish water and, in August, an infestation of little transparent jellyfish. The good side is that Coney has the Reigelmann Boardwalk, where you can stroll a mile back and forth, and partake of excellent American or Russian food, street food, the original Nathan's Hot Dogs -- that is to say, the original hot dogs! -- and fool around with bumper cars, shooting galeries and various rides.
Or take the A to Rock Rock, Rockaway Beach (make sure is the A to Far Rockaway, not the other A to JFK/Howard Beach). Mind you, the trip is well over an hour.
There are very few amenities, but Rockaway is a much better beach. This is an Atlantic beach. The water is clean, the waves are Alantic ocean surf: that is, long lines of big breakers. The surf is especially high if a hurricane is within a thousand miles or so. However, that means a rip tide, so bring something that floats.
If you don't mind walking a while, you can cross uot of the public beach to the Federal Recreation Area. Same beach, no people. Swimming isn't allowed, though.