First of all, Westchester, Fairfield, NEW Jersey, and Long Island (at least, not what New Yorkers mean when they say Long Island ) are NOT a part of NYC. You will find, when you come here, that there is a big difference between NYC and its suburbs. Second of all, Manhattan, The Bronx, Brooklyn, Queens, and Staten Island are not neighborhoods . They are boroughs. A borough is a political distinction for these five LARGE sections of New York City, each the size of a large city by itself. (Brooklyn is about the size of Philadelphia, for example!) Each borough is also a county, making NYC the only city which is made up of five separate counties! Within each borough are dozens and dozens of neighborhoods. For example, I live in the neighborhood of Park Slope in Brooklyn. It s a nice residential neighborhood, but fairly pricey these days. If you can afford to live in Manhattan, some VERY nice neighborhoods are the West Village, Soho, or Tribecca, but you probably can t come close to affording to live in any of these! Many people who can t afford Manhattan end up in Brooklyn or Queens these days, though less in Park Slope then they used to - Park Slope is almost as expensive as Manhattan today! But, if you are interested in Brooklyn, you might try Kensington, Prospect Heights or Windsor Terrace these days. Sunset Park is a real bargain, and I predict it will be the next up and coming neighborhood. Also, you might like DUMBO or Greenpoint, which are close to Williamsburg (considered a very hip neighborhood, but also way too expensive these days!) As for Queens, I know less about that, but you might try Astoria or Long Island City. And if they are too expensive, you might try Elmhurst. Good luck.
NY is the safest big city in America, but coming from the west coast, it will be a culture shock for you. Tons of community colleges, but NYC public schools have a hiring freeze in place right now if you are looking for a job there, also NY has VERY strict and high standards for schoolteachers. If I got a masters in teaching from California, I would still not be certified to teach in NY until I jumped through all their hoops. I live currently in the murder capital of NY. Hooray! Way worse here than NYC.
I live in NJ and it s very safe here. So NJ is the safest, then Manhattan (but very expensive), then Brooklyn, then Queens. I wouldn t recommend the others. Good luck!