Save up tons of money for the trip. You will need money for a plane ticket and money to support yourself until you find a job.
Buy your plane ticket at least a month in advance. The earlier you buy the ticket, the more likely you will find deals and non-stop flights.
Find a place to live. If you do not have any connections to New Zealand, you can always fly there and stay in a hostel until you find a room to rent.
Get a visa so you can work.
Fly to New Zealand in the beginning of the ski season so you can start searching for a job early. It will be harder to find a job in another country so you may have to be open to lots of possibilities. Work for the mountain to get a free pass and free transportation on the bus up to the mountain.
Bring a pair of skis from home. Ski equipment in New Zealand is generally more expensive than it is in the US.
Make sure you have a passport.