I think the only real difference between going for 24 months and 12 months in New Zealand is that for the 24 months visa you need to send in evidence of all kinds of medical tests. But yeah, no need to do seasonal work or anything to apply; it's a longer-term single visa, not a second additional visa which becomes available after you've done the first one.
It's been five years now since I applied for my IRD number (man, time flies), but I remember it being a relatively swift and painless process. Took maybe 1-2 weeks to arrive? I had family which I used for an address, but I know there was a travellers contact point as well which could organize it, and so using a hostel's address should be fine as well. As in Australia, the country is set up to deal with backpackers.
As for where to go in New Zealand: Auckland is the big smoke, with more than a quarter of the population living there. A lot of people in the rest of New Zealand see Auckland as being separate, but as a backpacker, it makes for a great base if you want to do office work and the like, and personalyl I found it a very pleasant city where there was always something going on.
Most flights from Australia fly into either Auckland or Christchurch, so those probably make for the most convenient starting points. With one or two years there, you'll get to see all of the country, so I won't give you too many recommendations, but my absolute favorite location ever in New Zealand (which I return to any chance I get (and rave about here on the forums only slightly more often))
is Lake Tekapo, halfway between Christchurch and Queenstown. It's a tiny village, with really not much to do but to do a couple of hikes and gaze at the color of the lake ...but man, what a color! Linger for a couple of days and watch the light do amazing things, different each day. Also make certain to visit the South Island (or at least the part of it on and south of the Christchurch-Queenstown line, as the west coast doesn't change much) at least once during each of summer, autumn and winter. Summer is the best time of the year to be there weather wise, but the autumn colors in that part of New Zealand are stunning, and the rare good days in winter are the most gorgeous of them all, with the Southern Alps all gorgeously covered with fresh snow.
As for Australia... a lot depends on where you've already been. Personally I'm quite partial to the area around Melbourne. The Grampians are great for hiking, the Great Ocean Road is a must do of course, and I love Cradle Mountain National Park in Tasmania.