Figure 30-40 minutes by train to Manhattan, if you are going to go there make sure you know where your going ahead of time.
NJ Transit from EWR to Penn Station Manhattan is about $11 one way, however if you have a few extra bucks I would recommend taking Amtrak from EWR to NY Penn (at least on the way into Manhattan). The reason is that you will be arriving at the peak of rush hour, finding a seat on a NJ Transit commuter train during rush hour is almost impossible.
It costs $20 on Amtrak one way but it's way more comforable (like First class airline seats) and you will be able to grab a seat and possibly a a bite to eat in the cafe car on the way.
My advice if your heading to Manhattan is to take Amtrak in and NJ Transit back, cost is about $31 dollars rt.
Here's EWR's rail schedules including Amtrak and NJ Transit .