The best way to get there is with the free ferry that leaves the southern tip of Manhattan. Some people just ride the ferry back and forth for the amazing views of the NYC harbor, Manhattan, Brooklyn, New Jersey, the Statue of Liberty, Ellis Island, Governor's Island, bridges connecting different islands… Like here, you see a bunch of helicopters against a backdrop of the Brooklyn Bridge and Manhattan Bridge:
And here's the Verrazano Bridge connecting Brooklyn and Staten Island (with a Manhattan-bound Staten Island ferry passing us by in the foreground):
You can't not take a photo of the Statue of Liberty too. It's a compulsion.
The ferry terminal on Staten Island conveniently has a bunch of bus terminals radiating off of it, with buses ready to take you to different parts of the island. This visit we stuck to the northeast part of the island, first taking the S40 bus to the Snug Harbor Cultural Center and Botanical Garden. Crossing the two-lane road from the S40 bus stop to the back entrance of Snug Harbor is a treat, because there are no traffic lights and crosswalk, and cars are zooming at you from both directions (in one direction hidden by a bend in the road). So you've got to time things just right and then hustle.