There is a large variety of options for travel from Newark Airport to Manhattan.
I have assumed you're travelling to the Port Authority Bus Terminal (as per the bounty), a short walk from Times Square.
I have listed them below in decreasing order of both convenience and price .
Take a direct taxi - $55 plus tip , tolls ($8-10), baggage fees and a $5 rush-hour fee.
Count on ~$70-$80 all in.
A shared van service, for example Supershuttle .
They pick you and a bunch of other passengers up and drop you each off at your own destination.
$23 each way Take the Newark Airport Express bus .
Departs directly outside each terminal on level 1. $16 one-way, or $28 round trip.
This goes straight to the aforementioned Port Authority Bus Terminal
The NJ Transit Train .
Follow signs to the AirTrain on arrival and buy a ticket to Penn station.
There is one change of train, from the airport monorail to a "proper" train.
$13 each way.
This lands you at New York Penn Station which is a 10-minute walk from the bus terminal or Times Square.
The following two solutions are included for completeness; I doubt they'd be suitable for yourself.
Take the #62 bus to Newark Penn Station ($1.60, exact change), and change for the NJ Transit Train (the same one as you'd take from the Airport, but for a lower fare) heading for Penn Station for $5.25.
Total cost of $6.85 .
Take the same bus to Newark Penn Station, buy a Pay-Per-Ride Metrocard with $10 credit on it from one of the PATH vending machines and use same to travel on the PATH train to World Trade Center, changing there for the Subway A or C train to "42 Street - Port Authority Bus Terminal" Station.
Note that this is a reusable card valid for the Subway and NYC buses, and $10 is exactly enough for a round trip from the Newark Penn Station to any New York Subway station.
You may wish to add more if you intend to use the Subway during your stay.
Total cost of $6.60 , not including the one-off card fee of $1.
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If your destination is lower Manhattan, you can take the New Jersey Transit bus to Newark Penn Station and then take the PATH train from there to the World Trade Center.
This is especially attractive if you have a relatively small amount of luggage, as the total fare is around $6, but it involves carrying your luggage around.
It is also possible to change to a different PATH train that will take you to Christopher Street or to any of four stations on Sixth Avenue between 9th and 33rd Streets.
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My experience was to be met by a stretched limo with a local driver (Mitch)
- I recommend it, best I know of.
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You have multiple options: By rail.
Read more about it on NJ Transit official website http://www.njtransit.com/rg/rg_servlet.srv?hdnPageAction=AirportConnectionsTo By bus.
I think it costs around $20-25.
You can always ask at Information Desk at the airport where the bus stops and how much is it.
By taxi.
They have flat fee.
Probably around $50-70 depends what part of Manhattan you are going to.
Don't forget tolls and tips.
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An economical and convenient way to get to the Port Authority bus terminal in Midtown Manhattan is to take the AirTrain Newark (a monorail) to the airport railway station, and from there catch an NJ Transit commuter train to New York Penn Station, from where it is a single subway stop (line A, C or E) to Port Authority terminal.
This route will take roughly 50 minutes and cost 15.75 dollars in total.
There are also direct buses from the airport to the Port Authority terminal, which cost USD 17.
However, be warned that congestions on this road, as in much of urban America, can be disastrous and lead to a very long travel time unless buses have dedicated lanes on the road from Newark to New York.
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I am a small town country girl traveling to NYC alone for the first time. I will be arriving at the Newark airport.   How do I get to Manhattan from there?    What is the best mode of transportation?
New York City from Newark airport
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<airport-transfer><new-york-city><ewr>
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There are many options, including a taxi ($55 plus tip, tolls, baggage fees and rush-hour fee), shuttle bus, #62 bus to Newark Penn Station ($1.60) and then switching to the NJ Transit Train to Penn Station ($5.25) or the PATH train to World Trade Center and then the Subway A or C train to "42 Street - Port Authority Bus Terminal". You an also take the AirTrain Newark (a monorail) to the airport railway station, and from there catch an NJ Transit commuter train to New York Penn Station. Finally, there are also  direct buses from the airport to the Port Authority terminal, which cost $17. The options for travelling from Newark Airport to Manhatten are: train, bus, taxi or shared van.
The options for travelling from Newark Airport to Manhatten are: train, bus, taxi or shared van.