Iam not aware of any particular phones or service in NZ.
I had my cell phone from Canada with me that I disengaged from my home network that prevented any roaming charges and any in-coming and out-going expensive phone calls. However, it did allow me to receive and send e-mails as long as I could connect with someone's wifi system. Most hostels, hotels and airports will provide you with a pswd for access. about 6 years ago
NZ - try "2 degrees" for cheap prepaid calls with good coverage. They use the Vodafone network. Telecom will generally have more coverage than VF, though not always. Depends where you are going, bro.
In Australia, if you are going rural you will NEED Telstra next g. Telstra charge like a wounded bull, but they get away with it because their coverage is incomparable. If you're staying around metropolitan areas, you have a huge range of cheap 3G options (Amaysim, boost, red bull, tpg, even Vodafone and Optus). You can buy a new prepaid phone for $50-$200, though these are generally all network locked. Jump on their websites to see rates etc. Try Dse.com.au or auspost.com.au or jb hi-fi for prices of new prepaid phones.
Maybe just buy a cheap unlocked phone and use it for both aus and nz. Depends if you want an iPhone/galaxy or can make do with a basic cellphone really. about 6 years ago
Unless a smaller provider has come along lately that I don't know about, Vodafone is the only provider that sells options for call time within NZ to non-New Zealanders. They make you buy a rather expensive SIM card, and then it's a prepay plan. Every other wireless dealer requires a contract for longer-term use, which generally also requires proof of residency. And call time is expensive, though NZ cell phones only charge for outgoing calls and texts, not incoming. Most people get the card and put it in their 3G or 4G phones, or find a cheap Vodafone brand phone (and possibly a Vodafone SIM card too) on www.trademe.co.nz. Then buy an international phone card to call long-distance. about 6 years ago