For Jewish Deli-style sandwiches...Katz's on Grand St. So good you could plotz! Reasonable prices.
Chinese food: Wo Hop (below street level down one flight) No. 15 Mott Street 212-566-3841...after 5:30 PM and you'll wait for a table. it's worth it. Off Mott St, you can buy 5 dumplings for $1.00 and they are great. Eat them while you walk. A great, cheap nosh. See Video.
Joe's Shangai: DELICIOUS soup dumplings, Dim Sum, etc. Family style dining. Good way to meet and greet New Yawkers. Try Salt and pepper Shrimp....delish! Reasonable prices.
Little Italy...over rated. Bring your money. Better yet, eat at somewhere close, and go there for Italian pastry and black coffee. China Town is literally across the street...Canal St. So, it's close.
Chelsea is not far, in fact, you could walk East, take in the sights, and then, chow-down on some good eats. Chelsea is loaded with great restaurants, for that matter, so is all of Lower Manhattan. Greenwich Village ain't too far from where you're staying, but it isn't what it used to be. It used to be "interesting", now, it's just blah, and sleezy.
Canal St is famous for designer knock-offs.
You'll love it! Enjoy the "Capital of the World"....The "Big Apple" You could stay awake 24 hrs. a day, and never run out of things to see, and do. Sleep only if you have to.
Frank