My friend, Amtrak is intercity, it does not provide commuter service in this part of the Northeast.
With that said, the best way to get from the airport to New York City (you didn't say which part of the city you're going) is by public transit.
Step 1: take the Newark Airtrain monorail from the airport terminal to the Newark-Liberty International Airport train station.
Step 2: at the Newark-Liberty International Airport train station, purchase a one way NJ Transit commuter rail ticket from EWR to Penn Station New York and take any New York City-bound (eastbound) train from there to New York Penn Station. A one way ticket on NJ Transit from EWR to New York Penn Station, which includes an access fee to and from the Airtrain monorail, will cost you $15.
Once in the city, you can take the subway or a taxi from Penn Station to your final destination. The fare on the subway is $2.25 per person (children under 44 inches ride for free), while the taxi fare varies by the meter rate with the initial fare at $2.50.
I hope this information was very helpful. If you have any questions or need travel directions while in town, just drop me a line by clicking on the avatar and e-mail icon in the profile. I'll be glad to help.
Good luck