It's "Times" - like the newspaper, not "Time's"
All you need to do is get yourself an unlimited daily metrocard and explore. If you don't want to look like a tool, never look up at the buildings. That's how you tell the tourists, they're the ones staring upwards. New Yorkers look down.
Most importantly, never stop at or near the top or bottom of a staircase, particularly in a subway station. And never block a sidewalk (stopping to take pictures, walking more than two abreast, etc.). New Yorkers are the nicest, most helpful people in the world, so long as you don't block one's way. Personally, so long as it's an able-bodied person, I have no qualms about knocking someone down a flight of stairs if they're blocking the way when a train I need to catch is in the station.