I led a couple student tours on tight budgets before moving to NY. Here's some stuff we did.
Museums and Parks:
The Alice in Wonderland Statue in Central Park
The Met - I think it's a must. It's huge. Be sure to make it to the roof. I highly recommend going on Friday evening. Bonus it's pay-what-you-want. (It's okay for each student to pay a $1, whatever they want.)
The Lower East Side/East Village/Little Italy/Chinatown (Best day of our trip):
Took a tour at The Tenement Museum (scheduled ahead for group tickets).
Shopped at Economy Candy
Walking tour with the East Village History Project (scheduled in advance).
Free time/lunch in Little Italy/Chinatown (Mulberry St., Mott St., Canal St.)
The Financial District/WTC:
St. Paul's Chapel - A mini Ground Zero museum/memorial that somehow survived 9/11 and was the makeshift relief center for first responders, free, near OneWTC, lots of history
Visited WTC memorial (this was actually before the memorial was finsihed, I had the students walk around, alone, reflecting)
Walked down Wall St., seen the bull, seen the Stock Exchange (from the outside)
The NY National Museum of the American Indian - In the the old U.S. Customs House, free, bonus not crowded, good a.c., and clean restrooms in a neighborhood hard to find those. I think the highlight is the contemporary art gallery.
Souvenir shopped at the NY City Store
Rode the Staten Island Ferry at sunset/the evening - Amazing views of Downtown Manhattan and the Statue of Liberty, fun picture taking, people watching, on a big boat, free
Touristy stuff I wish we did: