As a practical matter, start by picking up the map you see here, and studying both sides of it carefully (FWIW this map also neatly summarizes the rules for bringing your bike on the various forms of local mass transit).
- New York City just passed a new "bikes in buildings" law, which applies to commercial office buildings which have a freight elevator: if you routinely enter such buildings see http://www.nyc.gov/bikesinbuildings
- No amount of street markings or signage substitutes for paying attention to your surroundings: at the end of the day cabs do whatever benefits them, even if that involves intruding on a bike lane, and pedestrians are even worse (pedestrians randomly wandering out into the limited space I had between traffic to my left and parked cars to my right were my WORST problem today....).
- Riding a bicycle in Manhattan is an object lesson in self-assertiveness: if you have a valid traffic lane position or bicycle lane, you may actually be putting yourself at MORE risk by further weaving in and out of parked car clusters rather than just keeping a straight, visible, and predictable line in the space you have every right to take.
- On paper at least, NYC law requires your bicycle to have a bell.