If you're planning on using your existing phone on a NZ network with a NZ-based SIM you're pretty much limited to only Vodafone.
There's only two main mobile companies here, Vodafone and Telecom (smaller ones exist, but they're part of either and the same pricing plans generally). Telecom phones are CDMA meaning they don't use SIMs. One of the many reasons why I'm with Vodafone.
Telecom does have better coverage outside of the main areas, but it's only marginal, and really only a factor if you absolutly need to be in contact with the outside world all the time. All cities and the majority of small towns have Vodafone coverage.
You'd be fine with a Vodafone pre-pay pack, and I believe there's a place you can pick one up at at the Auckland Int. Airport arrivals area.