I trade with etrade Australia and also Direct broking in NZ.
google ANZ.co.nz to set your self up.
I dont bank with ANZ New Zealand but managed to obtain Access to the trading platform buy getting an ANZ creditcard, then I could open and link a trading Account.
However I find it is usually better to trade NZ shares that are asx listed on Etrade Australia
The NZ trading platforms are very substandard compared to what you are used to.