The Maori word for New Zealand is Aotearoa -- land of the long white cloud -- and it often lives up to that name. I hiked in NZ two winters ago (our winter/their summer). While the scenery is spectacular, the weather makes it hard to view. So make sure to waterproof your boots and pack your Gore-Tex boxer briefs; you'll need them.
On the North Island, I wanted to walk the Tongariro Crossing. When I arrived in National Park, my lodge hosts informed me that part of the trail was closed because of volcanic activity and that hiking the other part would have been futile thanks to low visibility and sideways rain. Fortunately, they steered me to other trails with somewhat better weather. This, by the way, is Mordor from the Lord of the Rings films.
On the South Island, I did some coastal walking around pretty Kaiukoura, which is the jumping off point for whale watches and open water dolphin swims. The dolphin swim was one of the highlights of my trip. (www.dolphinencounter.co.nz)
Then on to Arthur's Pass, which was beautiful. I walked the Avalanche Peak trail, which crosses spectacular alpine scenery. (And you'll get to see the Kea -- the world's only alpine parrot.) I had the best weather of the trip here.
I had planned to hike around the glaciers on the west coast (Franz Josef, Fox), but the road on the west side of the island were washed out. So be flexible in your plans.
The main reason you come out all this way is to experience one of the Great Walks. The Milford, the Routeburn, and the Kepler are the big three. Any of these hikes takes about three or four days to complete. You arrange to hike from hut to hut. The huts provide water and toilet facilities, communal kitchens and basic wooden bunks, but bring your own food, matches, utensils, and sleeping bag. The Milford Track was my first choice, but it was booked three months in advance. (Allow six months in advance if you want to hike there.) But I got my second choice -- The Routeburn track. (The less popular Kepler is probably easiest to book.) Whichever hike you do, first spend a day or two in Queenstown, where you can provision and arrange for equipment rental if you don't want to carry your own. You'll also need to visit the National Park office here to obtain your permits and receive weather updates. Queenstown is also the extreme sports capital of New Zealand, although most of the so-called sports (bungy, skydiving, luge) are just variations on ways to experience the force of gravity. And it's a party town. Everyone is here to have a good time. Tracknet provides transport between Q'town and the trailheads. It's a little pricey, but you don't have many other options. The Routeburn hike was wonderful -- alpine lakes, waterfalls -- but the surrounding mountains were often obscured by the variable weather.
See http://www.doc.govt.nz/parks-and-recreation/ for more info about the National Parks and Great Walks.
My hiking experience in Australia is limited to the Blue Mountains National Park, which is pretty to see. It's an easy day trip from Sydney to Katoomba, where you'll find many trailheads, though the hike around the Three Sisters formation is most popular. Australia is much drier than NZ, so fire is more a worry than rain.