The Red Bus is the local Roosevelt Island Operating Corp. bus that will transport you from the metro or the tram to various locations on the island. It costs 10 cents for seniors and disabled and 25 for others.
Once you get off of the Metro, you can use your metro card to transfer to a city bus to take you to the North end of the Island or even across to Astoria. Haven't taken it to Astoria yet so don't know where exactly it goes.
Getting to Roosevelt Island is easy. From Manhattan take the tram at the base of the 59th street bridge on the east side and pay your fare, a few bucks, sit back relax and take pictures. Or you may take the F train into Roosevelt Island. Make sure the train is heading towards Queens (uptown).
There are couple of choices to get to the island. First you can use the aerial tramway (which is exciting) starting at 59th Street & Second Avenue Station. The journey will take 4-5 minutes. The second choice is by subway, take F to Roosevelt Island stop. And of course there are possibilites to use a bus or your own car.