One thing I can tell u before u move. Buy some balls or equipments (shoes, clothes, and etc) from US and ship it or bring it over here.
prices of tennis equipments in NZ is generally nutcase expensive.
I used to live in US, so I found the price jump from there to here quite drastically.
I know for sure there are some free tennis courts available in Auckland area. One in mount Eden, very close to auckland cbd. But it's a hard court with an asphalt (the one that can use up your tennis shoes' sole). It's an outdoor.
Outdoor courts is quite common here in Auckland, even though the weather is not good...but for some reasons, Auckland is not having enough indoor court. I guess it's against the GREEN thing.
As for prices, I think indoor is $26/hr for members of the club and $16/hr for outdoor.
(I have to double check again).