I think the U.S. controlled a fair bit of the islands, but I bet Britain did too, and I'm not sure who had more.
Did you look up New Zealand on worldatlas? :-)
And as for the reason to colonize, think about why any country would want to expand into different areas of the world... whatever they would find there, they would own. What if they found things like gold, or diamonds, or oil?
And how did their colonization affect the indigenous people? Think about how those people lived before the colonization, and then after. Did things get better for them, or worse, and why?
I know I am giving you more questions here, instead of answers, but you're a good student! Learning how to "dig" will make you a better one!