From Morningside Heights go north up Broadway. This will take you through CCNY's campus on Harlem Heights (scene of a battle in the Revolution). Continue all the way to the hill at 181 Street. Many interesting ethnic sights. Enter the park on your left and walk up to the Cloisters, must beautiful spot in all NYC.
Down Broadway to 81 Street, then left to the Museum of Natural History (if you really like to walk this Museum will accomodate you!)
Going south, down Broadway to Columbus Circle (59 Street), see Lincoln Center, enter and go through Central Park.
You can do Greenwich Village and Chinatown in one day if you wish.
If you go up Broadway as far as 207 Street, turn right on 207, cross the bridge (where it becomes Fordham Road) and walk to the Bronx Zoo. This is a long distance walk from Morningside Heights, you might want to take the #1 subway to 207 Street and start the walk from there. This walk also gets you to the Bronx Botanical Garden.
Take the ferry to Staten Island, go left out the terminal on Bay Street. You can either walk Bay to van Briesen Park under the Verrazano Bridge, or turn left at Victory Boulevard and walk it as far as you like to view many Island communities.
Walk over the Brooklyn Bridge and explore Brooklyn
For Queens take the #7 subway to Flushing. Walk to the Bowne House Museum (a stop on the Underground Railroad), and explore from there.