Staten Island is easier to find an apartment in than Manhattan, but it still brings with it some of the apartment hunting difficulties people expect in NYC. In general, if you give yourself a month you should be able to find an apartment, but if you find that you're picky about accommodations, double that timespan to find something that you're truly happy with.
When you're renting an apartment in NYC, including Staten Island, you will find that many people recommend working with a broker. While you can find an apartment without one, you may notice that the search is easier if you move through a broker. When dealing with an apartment broker however, beware of high-fees that may greatly inflate the cost of rent compared to what you might find in a no-fee apartment.
As previously mentioned, driving on Staten Island is much more common than in the rest of NYC, and a car is a great way to get around the island itself. Although the bridge does provide a means of driving into Manhattan, most people find that to be more trouble than its worth, due to high traffic volume and the high cost of parking. There is a train on Staten Island, but it doesn't connect up with the NYC subway proper. Most people take the ferry to Manhattan, and then access the subway system.
Staten Island is part of the city that never sleeps, and although it is off in its own corner of the city, it is still part of NYC. Many people head into the city on weekends to take in the best concerts and theatre in the world, as well as to enjoy unparalleled dining and drinking. Summer nights on the boardwalk are a Staten Island tradition. Soon, you will also be able to visit the giant New York Wheel that soars 360 feet above the city.