Use this link and you can plan bus or rail depending on where you are in Carteret and where you wish to arrive in NY.
New Jersey Transit - Home
Rail is generally on time and adheres to a strict schedule. However, many people rely on the bus to NYC as well.
If you are going to lower Manhatten, you might consider taking rail to Newark, Penn station and changing to the PATH for the trip to NY. In fact, that option might save you some money.