Planning ahead is not a bad thing. Just make sure your plans are adaptable for when reality changes your perception, as it inevitably does.
Have you thought of travelling to the UK? If you live in NYC, it will cost you a lot less, take a lot less time, and give you the experience of being an American abroad. I like the idea of a gap year, don't get me wrong, but if you are intending to take on a lot of debt for your education, you might want to consider whether or not saving that money would be better.
Okay, nagging aside, check out a working holiday visa,
http://www.immigration.govt.nz/migra...idayscheme.htm. You'll see that this requires you to have 4200NZD. If you are willing to pick apples, sort hops, work in vineyards, there are a lot of jobs you could pick up with this visa depending on the time of year and try to not use that 4200. A good resource would be
http://www.wwoof.co.nz/
If you did a tourist trip to NZ, you'll need to research the cost of flights from your location, the cost of accommodation for the length of time you intend to stay (up to 3 months on tourist visa), the cost of what you need for nourishment, and the cost of travelling in NZ. All those costs will depend on how you travel and the exchange rate at the time of travel between the USD and NZD.
I'll leave the researching to you but here are some things to look into so you have an idea what to research:
Flights: Watch the various airlines to see when they offer best times of year for travel between your location and NZ. I have found that August and November usually have the best deals. August is winter and November is spring. You'll have to decide if it's a big difference in price at other times of year and if it's worth it for you to pay more to travel in the season you want.
Accommodation: Hostels and holiday parks are all over NZ and are relatively inexpensive for shared rooms.
http://www.bbh.co.nz/ https://www.yha.co.nz/
Food: It doesn't really have to be that pricey. Most hostels have shared kitchens so that you can cook up your own food and save a lot of money. You can have a look at costs at
http://www.countdown.co.nz/ to give you a general idea. Noodles and vegetables are a good basic backpacker feed.
Travel in NZ: To get from the North Island to the South Island you can fly AirNZ, Sounds Air or take the Interislander or Bluebridge ferries. There are various backpacker bus tour companies that you can jump on and off. I've not done them but I see KiwiExperience at the supermarket a few times a week. You can ask the hostels for various recommendations. You can also buy an old car and use it for travel to sell it at the end. That can be cheaper or more expensive depending on whether you are lucky and get a decent car or unlucky and get a clunker. Kiwirail goes from Picton (where ferries dock on South Island) to various southern locations. You can also try
http://transfercar.co.nz for cheap or free transfers if you are flexible and want to get from one place to another cheap and quick. Hostels often have ride shares.
You'll also want to look up the costs for activities you want to do.
You'll need to be 18 for all of this.
Good luck!