Not trying to be harsh, but if you have this kind of question, now probably isn't the best time to try riding in NYC. Your question is completely normal for someone just starting out--but putting yourself in heavy traffic sounds dangerous considering your skill level.
Btw, there's no set speed for taking a turn. Hypothetically you can take it as fast as the tires will let you. However, here are some tips for someone like you: grip the tank with your legs a little and let your upper body be loose. Don't pay attention to the racers draggin' knees right now. That's later on... As you enter the turn--Look where you want to go--turn your head into the turn. Look well ahead of your front tire--it will "slow things down" and allow you time to think about how to make the turn. Point your chest and lean your inside shoulder into the turn. You should be doing all of your braking (and then shifting down if needed)
before the turn. As you are in the turn don't give the throttle or brakes any abrupt input--be smooth. If conditions are bad (roads are wet or there's sand or something on the pavement) then be even smoother or try not to touch the controls at all. At your level you might feel like you're going too fast through a turn--you're probably not--and can lean a little more to get through it without running wide. If you are going so slow through the turn that you're worried about falling over, then give it just a tad more throttle. Use the clutch and throttle together to BE SMOOTH. Ok, so you're making your way out of the turn at this point, and you can begin to gently give more throttle. Did I mention to be smooth?
I'm assuming you basically understand the idea behind countersteering, so I didn't go into that--just the body motions that should go with it for negotiating a simple turn. There are lots of ways to take a turn--believe it or not--but the way I described is the way that I learned. It's easy and pretty intuitive.
You should really find someone to show you the ropes around the city before you venture off on your own, once you're ready. I live outside of San Francisco (much smaller and easier to ride in than NYC) and it took me months before I tried it. My first trip, I went in with a friend who knew which routes to take to keep me off of the steeper hills and away from the metro tracks n all. We went early on a summer Sunday morning to avoid as much traffic as we could.
And what Howie said--if you haven't taken the MSF, do so. It's a fun way to spend a weekend and you'll learn a lot! [thumbsup]