I find the easiest way to get from Newark airport to New York City, is the New Jersey transit train. It only costs $13 and a 25 minute journey.
There's signs all over Newark airport for the air train. Follow the air train signs and you'll not get lost, but there's always airport staff by the air train if you ever want help. The staff are always very friendly. One staff member offered to take one of my suitcases, but I like to do things on my own and I feel awkward when someone helps me. The air train stops at all the terminals and the airport train station.
The air train is free, but before you get the New Jersey transit train you need to remember to buy your train ticket at one of the machines. There's train ticket machines before you leave the terminal and in the building before heading out to the train. The platform is down a level, so you either have to use the stairs, escalator or lift and then wait on the right side for the train into New York.
There's only one train station stop in Manhattan and it's Penn station. There's a train station which sounds similar, Penn Newark and the Manhattan station they call Penn New York. I always check my map on my mobile, to see where I am. The other way to tell you're at Penn New York station, is everyone is getting of with suitcases haha.