Walk west along 42nd Street from the UN - see the Chrysler Building, Grand Central Station, New York Public Library, Times Square.
Take the subway under the east river (Train A or C) to the High Street Station in Brooklyn and then walk back to Manhattan over the Brooklyn Bridge. Awesome view of lower Manhattan.
Walk down Wall Street - past the Federal Building.
Go to Greenwich Village - wish it was 1962!
Take the Roosevelt Island Tramway (newly renovated!) - it's a cable car - across the East River (it's right next to the 59th. Street Bridge - feel groovy!)
Walk through Central Park - you could start at Columbus Circle: walk along the southern edge, walk up the east side through the zoo, and then walk west across the Park to the Dakota Building (where John & Yoko lived) and visit Strawberry Fields (the memorial to John just inside the Park near the Dakota).
Take the Staten Island Ferry (it's free!) from lower Manhattan to Staten Island. Then come back. It passes quite close to the Statue of Liberty.
Most of all, just walk, walk, walk!