Historically, it was possible to get to Long Beach Island by train, bus, or car. Now, the choice has been effectively limited to car. There is bus service, but they are limited, expensive, and/or complex.
By car
Long Beach Island is the perfect embodiment of American car culture. So it is almost certain you will be driving your own car to the island.
Take the Garden State Parkway to Exit 63. Then, take Route 72 E onto the Island. From there, you will want Long Beach Boulevard, which is the main street going the entire length of the inhabited parts of the island. There will be a toll at the exit.
-OR-
Take US Route 9 to the exit marked Route 72. Then, take Route 72 onto the Island. From there, you will want Long Beach Boulevard, which is the main street going the entire length of the inhabited parts of the island. As a US Highway, there is no toll.