From the Newark Liberty International (EWR) Airport, I wish to travel to 169th street, nearest to the Jamaica Centre MTA LIRR subway station on the 'F' line.
What are the possible rail transport options?
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Based on your specific request for rail, your trip will be a challenge if this is your first time traveling to the city, but it is possible. I have separated the answer from region.
Option 1. Take the NJTransit, Amtrak to the F Train
You start by looking for the AirTrain when you leave your plane. You can ask the airport employees if you are unable to find the sign. Once found, make sure you are on the platform headed to Newark Liberty Airport Station (which should be the last stop) Once you leave the train, look for an area that says NJTransit. Somewhere in the station should be a person in a ticketing booth that will sell you tickets. (Let them know that you want to go to New York Penn Station and ask where is the track. Once you find the train, stay on until you have reached Penn Station (most likely the last stop).
Here where it gets a bit tricky. You have to leave the building before you can get on the F train. Most likely from where ever you leave, it will be somewhere around 34-35th st between 7th and 8th Ave. Your goal is to walk to 34th St and 6th Ave. If you get lost, you can ask someone how to get to 34th st. From 34th st you should be able to see entrances that lead underground with solid colors with white letters in the middle (You should be able to see the F train symbol). To get pass the turnstile, you need a single ride ticket. You can buy this from one of the vending machines. You swipe your card through the reader on the right side of the turnstile and you walk/ push through. Once inside the station, look for the F train symbol on the overhead signs. Follow the sign until it leads you to an area that leads you downstairs to the platform. Make sure that you get on the Uptown platform. After you get on the right platform, look for the F train symbol overhead on the platform and wait until the F train comes. You should be able to identify the train by it's electronic symbol on the side of the train.
Option 2. Take the Airtrain, NJTransit, LIRR Nate brought up a good option to take the LIRR from Penn Station.
Here are some useful links to help you plan your trip
NJ Transit Ticket http://www.njtransit.com/ti/ti_servlet.srv?hdnPageAction=RailTicketsTo
Edit: Switched Amtrak transit to NJTransit thanks to Nate reminder of NJ trains.