My Recommendations:
Teariffic Cafe

The Teariffic Cafe has great bubble tea and though I have yet to try the food it is quite affordable and looks pretty safe compared to some other stuff in the area. A very modern looking place inside and out.Quite a popular place on the weekends, usually a younger crowd.
51 Mott St.
69 Bayard Restaurant
Just look for the neon "69" sign and the walls covered in dollar bills. This place is sort of like a Chinese Diner in that it provides quick food for a reasonable price and is one of the few places open late night in Chinatown. Although I have only tried a few dishes here I have not been disappointed. I suggest the General Tso's chicken because it is actually all white meat and not some shady brown chewy chunks of chicken that you might find at other places.
69 Bayard St.
Aji Ichiban and the "Unknown Candy Store"
Aji Ichiban as well as another candy store whose name I could not find that is right down the block are both an experience. They are both pretty much the same place. They have samples of just about everything for anyone to try with some of the items probably only for the daring such as the dried squid or the dried mini crab pieces.
37 Mott St.
Yunhong Chopsticks
I had no idea there was an entire store dedicated to chopsticks but this place is awesome. If you are in the area be sure to stop in here even if you do not plan on buying any chopsticks or even if you don't have a clue on how to use them.
50 Mott St.
Various Chinese Bakeries
With dozens of bakeries to choose from in Chinatown one may get confused on where to go. For the most part they all have similar items and I have yet to experience enough of these bakeries to be able to suggest favorites but a few of these in the area are pretty popular including Mei Lai Wah, Fay Da Bakery and Egg Custard King. I suggest getting a custard bun at least once in your life, all of these bakeries make them and I have yet to have a bad one.
Mei Lai Wah 64 Bayard St.
Fay Da Bakery 83 Mott St.
Egg Custard King Cafe 76 Mott St.