Well I can tell you what it's like here in NYC since I teach here. But I grew up right outside the city, so it was fairly easy for me to get a job. Usually locals have an edge over people who are trying to relocate.
The teaching job market is extremely tight right now. Schools are losing lots of money here in the public schools. Your prospects of finding a job greatly depend on what your certification would be. If you're certified in Science and Math, you'll have a better chance. A few years ago people used to say that if you had a pulse, you could find a job teaching in NYC public schools. Not anymore.
As far as the pay, it's very good. But still, even with the high cost of living, you can still find a decent place to live. I think the starting salary is 48,000 and that's just base without a Masters degree. You can also get $3400 a year if you work in a Title 1 school (most of them are either in parts of Brooklyn, the south Bronx, and Harlem). I'm a 2nd year teacher, on a Masters step, and I get the $3400--I make 58,000, which is MUCH higher than most areas of the country. I can afford a one bedroom or studio up to about 1800 or so. I have a 1 bedroom apartment in a nice/safe neighborhood in Queens, right by the river and a few minutes from Manhattan. At this salary I could probably find a studio in a decent neighborhood in Manhattan, but I'd rather have my cheaper apartment so I can save money.
The schools in NYC can be either very good or very bad. I teach in the south Bronx and my school is like a war zone. There have been 150 physical attacks on teachers from these (k-5) students from September-December. Many of these kids are violent and have very little respect for teachers/adults. It largely depends on the school though. If you have a supportive/strong administration, you wouldn't have the problems we have.
It's much more difficult to get a job in one of the decent schools in good/wealthy areas of the city. Nearly all teachers start out the first few years in a bad neighborhood, then work their way to better schools/neighborhoods.
It's still great living here. I love NY, born and raised here. I wouldn't live anywhere else (well, maybe when I retire, we'll see).
If you have any questions, message me