Kris- have fun, some of my favorite memories of my 20+ years in NYC were when my teenaged nephews came to visit. Seeing the city through a young fresh set of eyes is amazing.
The nephews rarely stray from steak and potatoes- yet they made me take them back to Ruby Foo's in Times Square 3 times. It is very accessible American/Asian food, served on platters made to be shared. If your group is large enough you sit at a huge round table with a lazy susan in the center so the food is easy to get at. It is on B'way at 47th St. and decorated in what I can only describe as Disney meets opium den. And only a five block stroll to Lion King. Outside the restaurant you can sit for a charcoal portrait for 10 or 20 bucks (the kids loved that for some reason.)
Other places that are good for kids on a budget (though not necessarily in the areas you mentioned)
The 2nd Ave Deli 162 E 33rd St. It is no longer on Second Ave but still has delicious cheap real diner/deli food. The servers are no longer rude and insulting which was part of the fun- but you will eat well.
The City Bakery 3 W. 18th- right off 6th Ave. It is a huge cafeteria where everything is made fresh and the offerings range from very safe to slightly edgy. IMHO the best chocolate chip cookies in NYC.
For lunch in Chinatown HSF 46 Bowery is insane fun- dim sum on trollies- enough egg rolls and sweet and sour to make any kid happy, and enough snails and duck feet passed around to make the kids realize that there is another world out there.
If you find yourself on the Upper West Side Big Nicks Burger and Pizza Joint is an institution. Always cheap- no frills at all, just good eats.
One site I recommend is:
http://www.nymag.com/agenda go to restaurants and then you can keep narrowing your search as to location and price, etc.
Again, have fun Kris.