Unfortunately (for you), many countries don't have credit card rewards markets that are as competitive as the US. The upside is that merchants are not ripped off by the credit card transaction fees that fund the rewards.

Several bank credit cards give points for Air New Zealand or Qantas flights (ANZ and Westpac are the main ones). American Express also has a card in New Zealand. These cards all have annual fees, so you'll have to figure out if you spend enough money at places that accept credit cards to make it worthwhile for you. If so, get the card and you may occasionally earn a free flight or two, but don't expect the types of massive sign-up bonuses and rewards that you could get in the US.