The best way for you to see New Zealand is to work and travel throughout the country with a Working Holiday visa. This type of visa is great because it allows you to legally work and travel in New Zealand for one year, so you can make money as you travel, instead of having to save up tens of thousands of dollars beforehand.
In fact, a Working Holiday Visa was how I was able to travel the majority of New Zealand. However, I did it without much of a plan. When I arrived I didn't have any idea where I was going to stay. My lack of planning lead me to some pretty great adventures, but it also caused me to have to look for work in some places where it wasn't worth my time to do so.