There are YHA hostels in someplaces which have very cheap accommodation, when i stayed in auckland it was $25 per night.
Wellington doesn't have that much to offer, but at the zoo you can feed a giraffe by going to the talk about them (so no extra charge to do it)
In auckland theres auckland zoo & kelly tarltons. I went there at the end of nov. The penguins were sitting on eggs (one had a baby that i got to see!) and they had carpet shark capsules, they stay in the capsule for 9 months, so they could possible be there when you go there. I did a zoom tour at auckland zoo (the cheetah one) and i'd fully recommend doing one of those.
New Plymouth doesn't have much there. But one thing thats near new plymouth is Stoney Oaks, i stayed the night there and i got to feed all the animals, & i could go in lots of the enclosures. its in Inglewood.
Hamilton zoo is definitly worth a visit