Re: NYC - places to go
You can probably ask the Front Desk at the hotel you stay in, about buses/trains but I think you can take the 40 bus from Elizabeth to Newark Penn station. From there you can easily board trains headed into the city.
First you would take the PATH train to 33rd Street. Then from 33rd Street you would board the subway for a train to 8th Avenue. There are little booths at the station and you can ask which train to take uptown. Here is a bus schedule.