There are two Hampton Inns in the city that are fairly inexpensive and also provide a basic breakfast--one is in midtown and the other is near the Brooklyn Bridge on Pearl Street. I recommend you spend at least a day walking around Greenwich Village, Soho and Tribeca. These are the neighborhoods below 14th Street, and have smaller buildings with lots of shops, apartments and restaurants that are true neighborhoods. The Upper West Side is very nice, too (between 72 and 90 Streets, west of Central Park along Broadway, Columbus & Amsterdam Avenues).
Of course, there are great museums like the Metropolitan and the Museum of Modern Art (and Natural History, if you like dinosaurs). If you like jazz, classical music and/or dance, there are dozens of performances every night.
I wouldn't go above 130th Street, or on the East Side between 14 - 34 Streets, because there isn't anything interesting to see there (compared to the rest of the city). Take a walk across the Brooklyn Bridge and a ride on the Staten Island Ferry for great views of lower Manhattan and the Statue of Liberty (the ferry goes right past it, very close).
Basically, I recommend just walking as much as you can in as many parts of the city as you can in the time you have. You will discover something wonderful on nearly every block!