Summer: Spend the day in Central Park or Battery Park City (look at the various activities beforehand, music, outdoors cinema) - As a tourist
Go to the beach (all New Yorkers do) - get a ticket on the Long Island Railroad and go to Jones Beach, Long Beach or Fire Island
Winter: Dress warmly, rent a bike and take a ride by the Husdon river toward Battery Park City.
On Saturday, go to visit many of the Chelsea galleries, followed by brunch
Any season: go to Chinatown for a Din Sum lunch
This tip has the address "Chinatown, Chelsea, Battery Park, Central Park"