If you didn't know before, you now know that public transportation is the best way to get around New York City. I'm not making this up: traffic in NYC is crazy. You can be stuck at the same traffic light for more than 10 minutes while driving through Times Square. Find out how to walk or drive step-by-step to your destination on mapquest directions
So, if you're going to New York City, learn how to use the subway here.
You can also take the bus, but I've always thought the subway was better. If you need to get to Staten Island quickly, you can take the Express bus or the free Staten Island Ferry, both of which I would recommend if you want to see great views of the skyline of Manhattan and the Statue of Liberty.
But if you don't want to take the subway, you can always hail a cab, just like in the movies. When you see a cab coming, stand in or near the street with your hand up.
You can also use Uber or Lyft. I've found that Uber/Lyft is cheaper. I've never had a problem with taxis running the meter, but it never hurts to check.
