'X' responded to this question in a very thorough manner so here it is.
Assuming you don't have a car:
As a young person, the few popular areas within commuting distance (by public transit) are: Hoboken, Jersey City (Downtown, Journal Square ["JSQ"], and the Heights), and Montclair. Maybe Harrison. With the exception of the Heights and Montclair, these areas all have access to Newark Penn via the PATH train. The Heights, you can bus/walk to the JSQ station or take the light rail to the Hoboken station. Harrison is a bit more residential/suburban in nature compared to the rest, as is Montclair. You can get from Montclair to Newark Penn via NJ Transit. All of these towns/areas afford you access to New York Penn as well.
There is a correlation between safety and cost, obviously. Hoboken, Downtown JC, and Montclair are the most safe and expensive, followed by the rest.
Assuming you do have a car:
Hoboken would be out of the equation. But, the rest would still be in play, as would the nicer suburban towns around Newark (Nutley, South Orange, etc.) Bayonne could be an option too; it's safe, but gritty and boring and traffic can be shit there. As always, the more expensive the apartment, the safer the area is.
All that said, I used to live in Downtown Jersey City and loved it. I would venture to guess that most going to Rutgers/Seton Hall live in Hoboken and DTJC. Hoboken, you would expect to pay around 1,500
minimum if living on your own; DTJC probably 1,300 minimum for a studio. The Heights is a nice option because it's right above Hoboken--you get all the nightlife, some pretty insane NYC views, all at a lower cost. I would imagine studios go for around 1,100 up there, but I remember seeing some for as low as 900. Safety starts to become an issue there though (stay as close to Palisade Ave as possible). I would rank JSQ after that personally--costs about the same as the Heights.
If you don't care about nightlife/being around young people as much, then Harrison might be a good option. Montclair has a younger scene, but can feel very suburban at times--most people head into the city for nightlife.
I would not recommend living in Newark. If you absolutely must, only live either in the area immediately surrounding Penn Station/Downtown or the Ironbound, as close to Penn as possible (it will still probably run you above 1,200/month living on your own--cost=safety). Do not live in any other area.
Let me know if you want some more specific information about particular areas/anticipated costs!
I hope 'X' doesn't mind me sharing
