Depends on what you like to eat, and how adventurous you are. The more willing you're, the better rewards you'll get. I promise.
Here is a list of places that are somewhat less intimidating and very authentic -
For Dim Sum : Sweet & Tart. 20 Mott St and 76 Mott. 20 Mott is slightly bigger.
Shanghai dumplings : Joe's Shanghai , 9 Pell St.
Noodle Soup : Big Wong 67 Mott. This place may scare you a bit, but their roast duck, congee and noodles are the best.
Everything : XO Kitchen. 148 Hester Street. This place has it all. Japanese snacks, Korean, dim sum, noodle.... etc.
And try bubble tea takeout at Ten Ren (also on Mott Street). Bubble tea is pretty much sweet ice tea, shake up, with topioca in the bottom. It's great!
My advice - order what others are eating. Point and ask your neighbor if necessary. Most don't take credit card, so have cash ready.
Enjoy!