Unless you go to the outer boroughs, not very close to Manhattan, you will likely NOT find an apartment you can afford on your own.
I live 25 minutes away from work, within a short walk of a subway station on a line that terminates a few stops after mine when heading away from "the city" (get used to that - natives never call it Manhattan in conversation, they refer to Manhattan below 96th Street, about three-quarters of the island, as "the city" - north of 96th is Harlem, and anywhere else in the city is considered the outer boroughs, with the people residing there derisively referred to as "B&Ts" because they either have to go through a tunnel or over a bridge to get to "the city").
The apartment I share with my fiancee is a little over $1,200 a month, and it's a one-bedroom. Add in a garage parking space and it's roughly $1,400. We pay separately for gas and electric, while water and heat are included in the rent. Cable television without any premium channels (other than some Russian-American TV stations) is around $120 or so for one converter box and one HD DVR.
That same apartment in "the city", and the garage costs, can be double or even triple in any neighborhood worth living in.
For your car, expect to pay somewhere over two grand a year in car insurance, perhaps even closer to three. Gas in the city tends to run over $4 a gallon for regular. The city streets are rough on a vehicle - I've had my struts replaced quite a few times - so expect maintenance costs to be high. Also, my car was broken into twice and vandalized twice. I disabled the remote entry system because the interior trunk release button shared the same circuit - the last time the car was broken into, nothing of any real value was taken because anything worth taking was in the trunk, which was too much trouble to access after all the time the thief spent just trying to get into the car itself. He tried to pry the door open enough to pop the latch (and did a great job of scratching the paint) but in the end just smashed the window on the driver's side. (You WILL want no-deductible window coverage on your car insurance.)
But there's a big plus to living in New York - you often don't even need a car to get around. There's easy access to subways and buses in most of the five boroughs, commuter trains if you're further out, taxis all over "the city" (and some in the rest of the city as well), and the Zipcar service lets you rent a car for a day or for just a few hours if you have a big shopping trip in mind. You might find it cheaper to sell your car, or just keep it parked most of the time.
Work can be a bit touch-and-go. Most entry-level employment won't pay enough to let you live on your own. If you have a trade skill or a college degree, that will help but it's still not a guarantee. Job hunt online first before taking the plunge, and try getting the New York Times on Sunday for the Classified Ads (yeah, it's archaic, but some people still use them).
New York can definitely be a challenging place to live - but there's no other place like it. The old expression is, "If it didn't come from New York, it had to go through here for approval first." This city has EVERYTHING you could imagine in a major metropolitan area and then some. Just don't expect it to be all that cheap - a night out at a movie theater (not even Broadway, but just a movie) for two can cost nearly fifty bucks with concessions. (New York has the most expensive movie and theater tickets in the country.)
