I have an iPhone 5S with Verizon and want to know the best way to have a mobile phone while there without incurring a massive Verizon bill. I would like to have data and voice, but I can settle for voice only and use wifi when it's free. Can I buy a SIM card and change it out or should I buy a prepaid phone for the 2 weeks or "rent" one from Verizon (again expensive probably)?
You have many options to get a travel SIM in NZ, make sure your phone is unlocked. Apple allows this so check with Verizon before you go.
To check if your phone can work on NZ networks, this page says the A1530 version of the iPhone 5s works with the carriers listed. You can find that info on the back of the phone
Vodafone NZ: 3GB / 200min / 200txt (49$) (it doesn't say anywhere on their page but I assume they sell nano SIM as they already sell iPhones)
2degrees has a prepaid option for data only (20$ for 1GB, 10$ for 300) but you need to unsubscribe before leaving otherwise it auto renews at the end of the month. They have nano SIMs for iPhones
Spark NZ (formerly NZ Telecom) also has prepaid monthly options (data only, or voice and text), but their terms are less clear than the others, you'll probably need to unsubscribe before leaving as with 2degrees.