Newark Airport to Princeton
Train: New Jersey Transit train to the Princeton Junction train station. When you arrive at Newark Airport, take the AirTrain from the airport terminal, which will drop you off at the New Jersey Transit Newark Airport train station. New Jersey Transit tickets can be purchased at the entrance to the AirTrain stations in each airport terminal, and the cost is approximately $18 each way. (Your destination station should be Princeton, rather than Princeton Junction.) Keep your ticket as you get off at Princeton Junction, and from there take the shuttle (Dinky) over to the town of Princeton.
When using the online train schedule, please refer to the schedule for the Northeast Corridor Line. Here is a link to online information for the AirTrain and New Jersey Transit.
JFK Airport to Princeton:
Train: Take the AirTrain to the Jamaica Station (red line) then pick up the LLRR line (Long Island Railroad) (gray line) to Manhattan (Penn Station), from Penn Station you take New Jersey Transit Northeast Corridor Line to Princeton. (Your destination station should be Princeton, rather than Princeton Junction.) Keep your ticket as you get off at Princeton Junction, and from there switch over to the shuttle (Dinky), which will take you to the town of Princeton.