I know this is an old post but for anyone else interested, or if you ever go back, I have some recs!(this is for the ea. village/lower east side mostly)!
Joe's Pizza is a classic, Lombardi's and Cello's as well. For bagels there's Leon's, Tompkins, and Pop Ups. Ferrara Bakery in Little Italy is (I think) one of the oldest bakeries in NYC and they're amazing!
There are also so many community gardens in lower Manhattan, and they're free! So if you ever want a chill place to just sit those are always great. NoLita and Bowery areas have so many cute shops and little places to stop in (beware, some of them are P R I C E Y)!