How much money do I need to move out. I want to live in North Carolina by myself in a one bedroom apartment. Right now I live in New York.?

Probably 6000-8000. Do you have a job lined up in NC? If not, you'll have a very hard time, maybe not possible, renting a place.

Just like in NY location matters. Plan on at least $600+/month in rent plus your budget for the rest of your expenses which will be similar to what your parents pay for in NY. Talk to them about all their expenses, doing this on your own will likely cause you to forget many because you take them for granted. For example laundry expenses is something people often forget about, as well as utility deposits or installation.

Understand its not just about how much money you have, because landlord will also require you to have a source of income before they will rent to you and in most places will demand you make 3 times the months rent each month. Of course you will need to pay the deposit and first months rent before you can even move in too.

My advice is you build a budget using this criteria. You can find this by going down to the city you want to move to and looking around, finding a grocery store, a gas station and a neighborhood. Also, try finding a rent-controlled apartment. Big plus. Anyways, here are the things you should have in your budget:

Utilities, bills and taxes (different states and areas have different tax rates and you might need to have different bills depending on where you live; be very thorough in this)
Food
Phone
Internet (might not be needed if you get a data plan with unlimited hotspot/data, which I recommend in some situations)
Car./travel
Gas
Rent
Entertainment (if your budget is way too tight, cut this out entirely)
Gifts (Mother's Day, Father's Day, birthdays, Christmas, girlfriends, stuff like that)
Job expenses
Cable (like entertainment, a luxury that you can cut)
Insurance (VERY IMPORTANT)
Clothes (go to thrift stores and outlet malls and stuff until you get a full-time, well-paying job)

Once you have all of this, save up for the first six to nine months and set up a job, then go. If you need more help, comment and I'll keep looking back for a bit.

Also, might I ask, why NC? And where in the state? Also, keep in mind, not everywhere is like NYC, especially in the south and such. Not everywhere is gonna be more apartment-oriented, and smaller cities and towns hardly have any apartments at all, much less quality ones. If you want to move to these areas, I suggest a simple condo for an alternative.

A nice apartment is around 400$ to 600$, you can find for as compared to New York would be 850$+... You could just search them online, maybe location, price, call them whatever it is ask, etc...If you trying to Move. Your cheapest option would be renting a Truck or trailer. Uhaul has an option between States!....Unless You have a Car appropriate vehicle for "moving".. But if you Drive it shouldn't take more than a day! So all together with Renting a means of transportation, gas, food, Tolls, day of driving....should cost around 550$, with more stops, hotels, could be extra ...Then once you get there You Must have at least enough for 2 month rent! so around 1k$ So all together just to get there, move your things, and move into an apartment would be $1500 or so depending on your means! After that You must have Extra money to sustain yourself! $500 for a month on a safe side!
In total you need $2000 of hard earned money! To Move to another State, and Get a living place!