Amy, if you look down, you'll see concrete. If you look to the front, back, left, or right, you'll see buildings. If you look up, you'll see sky. And crowds everywhere. (I'm 6'3", so they're like bloody hobbits underfoot.) And the occasional smell of pee. Look for signs that say 'Dirk was here,' especially near the pee. Re-watch Blade Runner before you go.
Seriously, there are two very nice cemeteries downtown, one with tombstones dating back to the 1600s. The World Trade Center, of course (I never saw the rebuilt version, unfortunately). The NYSE on Wall Street with lots of security. Statue of Liberty & Ellis Island boat tours leaving from the very south tip, Battery Park City downtown (lived there once), and the walkway up the whole West Side of Manhattan (especially nice downtown in spring), with views of NJ. Tour boats around the island (a must, IMO, right under the bridges). If you like the stink of fish, go to the Lower East Side. 42nd Street (unbelievable crowds, hold onto your kids). Central Park. Broadway shows. A zoo in Queens, I believe. There's a botanical garden somewhere, too. I don't recall anything in Brooklyn, but I never went very far there.
And on day two, .... :-)