If you came to New Zealand and did all your training here, you would have the same chance as any other resident when it comes to getting a job - most operators are not prejudice. From what I've heard the same can not be said for many Australian companies with some of them even reluctant to take on kiwis.
The company I fly for at the moment (small regional airline) has pilots from many different backgrounds (about 20% of new recruits over the past 6 months have been aussies).
As mattyj said, you would have to work your way through the NZ GA seen to build hours, and that can take anywhere from 2-5+ years depending on the shape of the industry at the time, and to a large extent, your level of planning, and luck.
There are several good flying schools who could provide you info about training (keep in mind they are biased and may try to paint a slightly rosier picture of employment oportunaties). Give these ones a go to start with. (They all have websites).
Ardmore Flying School (Auckland)
International Aviation Academy of NZ (Christchurch)
Nelson Aviation College
Bay Flight International (Tauranga)
There are also many small aero clubs around the country who are quite good, and offer a different style of training.
Wakatipu Aero Club (Queenstown)
Waikato Aero Club (Hamilton)
Tauranga Aero Club
Hope that helps. Buy the way, NZ is a great place to live
