Yep, the NYC cycling map is excellent.
The best advice I've ever received regarding urban riding is to ride as if you are invisible. Always always keep that in mind, and never make any assumptions about what cars are going to do. Just because that car has a left blinker on doesn't mean they aren't going to turn right and mow you over.
NYC has always been a challenging place to ride, but Mayor Bloomberg has made the city a much safer place for cyclists. Stick the routes on the map at first, and you should be fine.