Hi there!
There is a lot of information about moving to the city, including neighborhood mapping/pricing to find a neighborhood that matches your price to comfort ratio. As other people have suggested, check out the NYC oriented subreddits.
I'm a native, so I can offer you some great information if you want to type up some more specific questions and post below.
As a first time arrival I suggest you look in Harlem or "SoHa" (thats south Harlem that they're trying to re-market) if you're comfortable in safe, black neighborhoods. Hamilton Heights, and Washington Heights if you're comfortable with safe, mixed neighborhoods. Basically the western and center top of the island. El Barrio, also known as East Harlem, or Spanish Harlem, can be a little scarey for new comers, and is rather notorious for its Po-Jos (Project Housing) so I don't suggest moving there unless you're confident, or are actively looking for a Spanish dominated neighborhood, or want to save money desperately.
Avoid the Bronx. The rule with brooklyn is the farther out you go, the cheaper the rent. WBurg will kill your wallet, both with rent and entertainment. Greenpoint has become pricey as of late, and E. bushwick is become the same, though there are mixed prices. Crown Heights has had a lot of recent construction and sometimes a good deal can be found, especially for short term year long deals as people try to find ways to pick up the new condos. Certain apartment complexes on second ave (for me it was the Normandy on 94th st and 3rd) may offer reduced rates in an attempt to fill vacancies while the 2nd avenue subway is being installed— but that comes with the caveat of early morning/weekend construction, which can be very loud. If you're really game to save money, you can also try shared space living, which is like 7-15 people living together in converted factory building floors. They're rather selective, and come with all the perils of communal living. An example of this (though now priced too high, and partially vacated) would be the McKibbin Lofts. TBH, if you're a guy you probably won't get into one, sorry.
Either way you'll definitely want roommates, and if you can try to find a rent-stabalized or rent-cotrolled building. Aim for $900 a month, hope for lower, but don't be surprised if its as high as $1200 or more. Almost every apartment will require first month and deposit, some will require last month as well. Heat, hot water are almost always included. Ask your building if the hot water is generated by the building, or in individual units, as older buildings sometimes have apartment water heaters that will kill your electricity bill— though its rather rare.
Food prices are above national average. If find a friend with a costco card or the like, try to do some bulk purchasing— aim for ingredients and necessities suggested on different frugal subreddits.
Monthly metrocards > buying as needed except when your transportation usage is low. People did the math on here, look through some of the post, but 99% of the time its better to just pay the $116.50. Expect train delays, we're rather angry at the MTA right now. ;)
I love me my A/D train when I lived in Harlem. It skips between 59th-125th making my cross manhattan commute about 20 minutes one way. If you live on the 6 train, prepare for madness and delays.
Uber sometimes has a discounted "first time user" deal that can get your one free uber ride. This is particularly useful for first timers who might get lost in BK and need a ride back to manhattan (~$15-30)
Biking is great here if you're a really good biker. You should practice a lot first. Right now is "pot-hole" season with all the winter damage and it is more dangerous than normal to bike. That being said, biking here is safer than it has ever been.
Your Rent: 12,000-15,000 yearly
Your Food: 2,400 yearly.
Your NYC experience: Priceless expensive.