Another native NY-er weighing in. If you are a foody I'd say determine what you'd like to spend on one really incredible meal and then check out the website
www.timeout.com/ny where they do a better job of having real reviews than the Zagat which I know most people swear by. That goes for all aspects of things to do in the city.
Sites mean different things to different people but if you are not into crowds do the touristy things early in the morning.
The Empire State Building does a great tour via audio program. They give a lot of history of the neighborhoods you are viewing from above.
I agree that the chinatown/Little Italy area is a great stop but you may not find the best restaurants just by walking around and stopping in. Again I'd check out a guide and have some place in mind.
Staying inexpensively is not impossible but again that's relative and means different prices to different travelers. I have stayed in corporate apts for better rates than hotel rooms and there is always a kitchen in them so don't have to eat out at every meal unless you want to. I suggest a google search for that.
One of the most beautiful churches is The Cathedral of St. John the Divine. it's uptown on 113th I think and Amsterdam. Worth the trip and on Amsterdam Ave around 109th is an old world Polish bakery which is wonderful for an after tour coffee and pastry break.
St. Patrick's Cathedral is the grande dame of NYs churches but there are so many lovely historic religious buildings throughout the city.
Avoid all tourist traps such as can be found in Times Square when looking for a place to eat. Unless of course you know what you are in for and want the experience more than the food. They represent the chain feel not the feel of such a varied city.
Have a great time and let us know what you loved and didn't when you get back.