We are planning to be in New Zealand for 10 months starting in August and currently are under contract with Verizon here in the USA.
I am thinking about suspending my Verizon account and getting local service in New Zealand. The other option would be to switch to T-Mobile that has global roaming data.
Will our iPhones work in New Zealand? Data is probably the most important part for us.
This is very much related to your phone and to a certain extent your provider. It is especially important for Verizon customers because their network is different to most of the world (CDMA vs. GSM) and hence the phones they sold often only worked on their network, unless they were marketed as 'world phones'. You should also consider the frequencies supported (Verizon and T-Mobile use different frequency bands to most of the world).
In this case though, the iPhone 5s is a 'world phone'. Looking at the specs (http://www.apple.com/iphone-5s/specs/),
Model A1533 (GSM): UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)
Model A1533 (CDMA): CDMA EV-DO Rev. A and Rev. B (800, 1700/2100, 1900, 2100 MHz); UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)
Model A1453: CDMA EV-DO Rev. A and Rev. B (800, 1700/2100, 1900, 2100 MHz); UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 18, 19, 20, 25, 26)
Model A1457: UMTS/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 5, 7, 8, 20)
Model A1530: UMTS/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); FDD-LTE (Bands 1, 2, 3, 5, 7, 8, 20); TD-LTE (Bands 38, 39, 40)
... you can see that the CDMA models (you should be able to confirm this in Settings>General>About>Model) support UMTS/HSPA+ (the 3G technologies in use in NZ) and GSM/EDGE (the 2G technologies in use in NZ).
The frequencies in use in NZ are -
2 Degrees (http://en.wikipedia.org/wiki/2degrees#Network) -
3G (UMTS) - 2100 Mhz (urban) and 900 Mhz (rural)
2G (GSM/EDGE) - 900 Mhz and 1800 Mhz
Vodafone (http://en.wikipedia.org/wiki/Vodafone_New_Zealand#Coverage) -
4G (LTE) - 1800 Mhz
3G (UMTS) - 2100 Mhz (urban) and 900 Mhz (rural)
2G (GSM/EDGE) - 900 Mhz and 1800 Mhz
Telecom NZ (http://en.wikipedia.org/wiki/XT_Mobile_Network) -
4G (LTE) - 1800 Mhz
3G (UMTS/HSPA+) 2100 Mhz (urban) and 850 Mhz (rural)
The frequencies match up to the ones supported on your iPhone, so therefore you should be fine on any of the mobile networks and you should get at least 3G data performance in urban areas, if not 4G. You might fall back to 2G on some networks outside urban areas.
Verizon has also previously stated that they do not carrier-lock the SIM slot on iPhones, so you should be free to use other SIM cards abroad (see http://9to5mac.com/2012/09/24/verizon-iphone-5-wont-ever-be-locked-most-likely-due-to-the-fcc/, but do check with them first though).
Finally, do consider the T-Mobile global roaming option, however I believe that only allows you to roam for 3 out of the previous 6 months or something like that, to prevent people from signing up and permanently using it abroad. I don't have any recommendations as to which network you should sign up for, but note that the 2 degrees network is only available within urban areas; outside urban areas, it falls back on to the Vodafone network and you may lose data access (although calling and SMS should be fine).
Since your phone is unlocked, I would recommend buying a prepaid SIM card in NZ, from either Telecom or Vodafone, as they are the largest providers and have the most coverage across the country. Since you will be there for less than a year, buying prepaid will likely be cheaper overall, and it is much more common to have a prepaid SIM in NZ than in the US.
There are many Telecom and Vodafone stores throughout NZ so locating one should not be a problem.
Telecom pricing: http://www.telecom.co.nz/shop/mobile/mobile/plansandpricing#ppPrepaid
Vodafone pricing: http://www.vodafone.co.nz/prepay/
