The problem with living in Jersey is you get socked with two state income taxes if you work in New York. Not great for a recent grad.
Here's a list of the NYC real estate brokerages. They also do apartments. The biggest tend to only handle upper market offerings but I recommend looking through them anyway to see what you're getting.
Frequently Asked Questions about renting in New York
Guidelines for Renting in New York
Be aware that this is a city where people read the obituaries to find good apartments. When it comes to apartments, brokers always work for the landlord. It's extremely rare to have a buyer's broker. The brokers will attempt to convince you that this rat and roach infested shithole with blood-stained walls in a 15 storey walk-up is the greatest find in the city. Don't let them sway you.
Apartments near the subway are best because it means you can pretty much get anywhere. Subsequently, apartments near the subway tend to go for more than the same apartment elsewhere. Most people your age do not really live in their apartments. They're out and about all the time. They use the apartment to eat, sleep, and poop in. That's about it (largely because that's all there's room to do).
This is
the subway map for NYC.
This is a
crime frequency map for NYC.
You do not want to live in the high crime areas and do want to live near a subway stop (preferably one that the trains stop at 24-7). NYC's subway runs 24 hours a day, every day, but after hours some stops are skipped. If you're out partying until all hours then you want a train that will get you home.
New York's apartment market is fucked because of what are called, "rent control," and, "rent stabilization." Ages ago the city thought it would be a good idea to put a cap on apartment rental prices and the only way rent in those apartments can rise is by approval from the city. This means there are some people living in Fifth Avenue apartments resembling a palace for $1400 a month, and other people living in shoeboxes for twice the price. About 40% of all NYC's apartments are controlled in some form and naturally, people don't want to leave these bargain places. My sister has a 2-bedroom apartment on East 97th and Third and pays $800 a month. She's one of the lucky ones. The result of all this is that there is artificially high demand for apartments because so many are technically off the market and landlords have to make money off the apartments which aren't controlled because they're losing money on the apartments which are controlled. It's messy.
Do you know where your employer is located? If not, find out. It's nice to not have to make a lot of transfers to get to where you're going. And buses suck. Yes they work, but in Manhattan sometimes traffic is so bad that nobody goes anywhere for a while. If you're on a bus, you don't go anywhere either. Subways do not have this problem.
It's rare to find an apartment with laundry facilities. Don't overlook the benefit of having laundry machines in the building. If they're not in the building, see how far they are from where you live. Same with groceries. You also want to consider how many floors up you live. Can you get upstairs if the elevator dies. Is there only one elevator? If there's no elevator can you get upstairs if you injure yourself? How will you get furniture up? Be aware of what's below and across from your windows. If they're closed, open then and take a few deep breaths. Is the taxidermy shop downstairs causing your apartment to smell like carrion? Like Chinese food? Is the back lot a community dump? Is heat and hot water included? Can the electrical system handle modern applicances? Can you get an air conditioner in? Do the outlets near the windows have 3-pronged outlets. Some old apartments don't have them. You will definitely need one so plan on buying at least one cheap model. Learn to figure how much air conditioner power you will need to cool the space you have. How will you get the air conditioner up the stairs?
There is a LOT to consider when renting a place you can live in and live with. There are some good NYC renting guides online from people who have gone through the process and learned the hard way. Read them and learn!