I used to live in brisbane, the Gold Coast is very touristy, theres all the theme parks to go to if you're into that sort of thing (buy tickets in advance, you can get a 3 or 4 park pass, MUCH cheaper)
On the way from Bris to Gold Coast, there are a few nice nature walks in the glasshouse mountains.
The sunshine coast is also quite touristy, but not quite so much as the GC, there are beautiful beaches and national parks.
Look up Currumbin Wildlife Sanctuary and of course Australia zoo - they're great places to visit.
In Brisbane city, the southbank markets are lots of fun on a FRIDAY night.\
I was born in NZ, dont go to christchurch, its still having earthquakes rather regularly, but in the south island, everything south of Dunedin is beautiful - scenery (mountains, lakes, snow) and Queenstown is the tourist destination for snow sports and adventure sports (white water rafting, bungee jumping). Google some tourist sites for info on the best seasons to go.
Im not that familiar with the North Island, except for tourist spots in Rotorua, and Waitomo caves.
Again, just google and you'll get a better idea of places and the best seasons to go.
NZ is still not the warmest down south, even in summer months.
Brisbane is "sub-tropical", so even if you go in winter, its not very cold, and quite humid.
Enjoy planning your trip and have a wonderful time!!