Wow! I'm so sorry I'm so late on this post! I was actually on holiday, and IN New York when you posted, so I wasn't able to see this... apologies!
Here's some general information that some have mentioned, and some other information that hasn't been mentioned yet.
There are a number of different Korea Towns in New York, all with their different nuances and specialties.
K-Town in Manhattan
(34th, between 5th and 6th) - Absolutely fantastic food. Real Authentic food that makes everyone happy. It's only one block (plus a few surrounding blocks), but it has everything a K-Towner would want (restaurants, 노래방, Korean supermarket, cafes and trendy Korean bars). It's a bit expensive to hang out there, but it's definitely Korean (but it leaves a heavy New York/Manhattan flavor).
You can take almost any uptown/downtown (almost all stop at 34th). And you can walk across to 5th and 6th.
some spots are "Korean." It used to be a heavy Korean town, but now it's become more mixed. But there are still spots in flushing that are heavily Korean. Namely "Northern Boulevard. Between Parsons Boulevard all the way to Bell Boulevard (which is in Bayside)" There are Korean spots everywhere along this area.
Along this, one of the best supermarkets you can go to near 150th St (on Northern Boulevard) is 한양. It has all the Korean food you need, but one of the best parts about it is the supermarket has a little 포장마차 inside of it. Great food, and it has a VERY Korean feel to it.
From Manhattan, you can get to Flushing via the 7 train. The last stop (Main St.)
Fort Lee, NJ
This area of New Jersey is less than an hour outside of Manhattan and can be accessed quite easily. It's mostly a residential area, and it has quite a large Korean population.
For all of these areas, I recommend 한아름 Supermarket. It's fantastic.
Also, there's Red Mango and Pink Berry, which are Korean yogurt chains, but have made it's way to the mainstream market.
For Fort Lee, there are a number of buses that you can take from Port Authority (7, A,C,E,1,2,3...).
All of these have Korean churches if you're looking for them.
I hope this helps you out, and please don't hesitate to let me know if you're looking for something more specifically!