ok....I will throw my 2 cents in, as a native NYer, and someone who still lives and works in NYC. My recommendation is to find a hotel in Williamsburg (Brooklyn), or Long Island City (Queens). Will they be cheap....no. But they're literally a stone's throw from Manhattan, as they're both right across the East River, and will have killer views of Manhattan. And Williamsburg and LIC are pretty hip neighborhoods themselves, with lots of great local restaurants, bars, art galleries, and quite a good mix of counter-culture at it's best. To me, these local neighborhoods come pretty close to representing the real NY for NYers, and less of the touristy commercial stuff. (which is fine too...after all...you'd be a tourist). These neighborhoods are both literally a 5 minute subway ride into Manhattan.
And then, once in Manhattan, my recommendation would be to walk, walk, walk, since that is really the best way to see the city. If you've never been to Times Sq, it'll be a place you should go see...and then probably never go back. My recommendations would be the East and West Village, Central Park, and to check out the progress that has finally been made at Ground Zero. It is solemn, hopeful, and a million other emotions, all at once.
Hope this helps....
-Chris
ADDITIONAL INFO:
Food recommendations:
PIZZA: You have to get some real pizza. Let me repeat: You HAVE to get some real pizza. A few years back...I went on a little quest for NY's best pizza. Having been here for 30 years...I feel I am qualified in this area.
#1 - By FAR the best I have ever had - DiFara Pizza, Ave J in Midwood, Brooklyn. (about 30-35 minute train ride from Times Sq). No words can describe what you'll experience, so I won't even try.
#2 - either Grimaldi's (under the Brooklyn Bridge on the Bklyn side), or the original Patsy's in East Harlem, 1st Ave and 118th St. Both excellent in they're own right, but #1 is in a league of their own.
BURGERS: I'd recommend "Shake Shack" in Manhattan, or the original "Jackson Hole", off the Grand Central Pkwy in Queens
I'll stop here....I am making myself WAAAAY too hungry. Best of luck.