If I was you I would come up to Auckland and do the Zookeeper course at Unitec. Its once year long, and then you can get a job at pretty much any zoo in NZ 9though you will have to start from the bottom and work your way up). Other option if you want to work with animals include getting a job in the biodiversity or pest management sectors of DoC, again you will need to do degree or the DoC ranger certificate (in Nelson if I remember correctly). Other options are pet shops, vets, public aquariums or animal shelters.
Good websites to keep an eye on include
http://www.conjobs.co.nz/
http://www.doc.govt.nz/templates/page.aspx?id=42183
http://www.conservationvolunteers.org.nz/ (although these are volunteer positions, they will help you get work experience and ultimately, a paid job)
Sorry, but I come from the conservation/zoo side of the working with animals, so cant really inform you if you want to get into the vet/animal shelter side of things