I lived in NZ for 18 years, and honestly, there’s bugger all of interest (in the North Island) outside Rotorua. Auckland has to be the most boring major city on the planet, and while Northland is very nice, there’s nothing there that isn’t in Australia.
The South Island is lovely, though- mountains, lakes, scenery, and lots of arty, crafty, outdoory things to do. Make sure you take the TransAlpine Express from Christchurch to Greymouth (and back!)- it’s one of the world’s premier rail journeys and has the most breathtaking scenery.
As for Australia? There’s miles and miles of bugger all between the major population centres, but Sydney and Melbourne are must-see places.
Brisbane isn’t very exciting, but Queensland itself has lots of palm trees and beaches, and you really should try and see the Outback if you can- there are trains running from Brisbane to various Outback destinations that have won awards and are supposed to be excellent.
I haven’t been to Tasmania yet, but from what I’ve seen it’s almost exactly like the South Island of NZ, so you can probably skip it.
Bear in mind getting around Australia and NZ is expensive- internal flights in NZ cost a fortune, and there are massive distances to cover in Australia which necessitate flying.
Still, I’m sure you’ll love it here! They’re certainly unlike anything in the US…
While I remember, check out Bill Bryson’s book Down Under A great read, and a good preparation for a trip to Australia.