Welcome to NYC!!! Please remember that NYC is huge and you can never go to "just the right place" – besides that "it" often changes so be sure a place that is recommended is still open,
As a Noo Yawka, I think the "NYC" experience is just being here, seeing whatever and whoever, and being open to eating whatever too. I like the idea of street carts becuase they are "real" and you don’t waste 1 or more hours of time sitting inside when so much of the NY action is outside! Just walk on the streets, go to the parks, find out some free or inexpensive NY Sumemr entertainment (concerts, etc. in parks) and enjoy! and come back for more (then you can give us YOUR NYC roadfood discoveries!