If you really want to get off the beaten track in Australian and New Zealand you will need to rent your own transportation since due to the spread our nature of the countries public transportation is pretty sparse and focused on the main centres and tourist areas.
I've done a trip from Brisbane to Cairns a few times and Arsenal has also done a trip from Sydney to Brisbane. The easiest place to focus your travel is along the east coast working your way up and down and there is plenty of stuff to see there, and you can easily make excursions further into Australia.
Australia's biggest drawcard is its wildlife. I'd be sure to visit the rainforest areas of northern Queensland and you are bound to see all sorts of animals from crocs to cassowaries. There is a nice hostel set in the rainforest at Cape Tribulation that is worth visiting. Also Broken River if you want to see wild platypus. Magnetic Island is also a nice place to visit and they have a wildlife sanctuary where they let you hold snakes and baby crocs. Apart from that the main cultural centres are Sydney and Melbourne, although I haven't been to Melbourne.
As for New Zealand it is best tackled with a trip from North to South. The beaches and forests of Northland are worth exploring and you have to dry dune boarding in the Hokianga. Also should visit Rotorua and Taupo for the geothermal activity, think Yellowstone. Wellington was recently voted the best capital city in the world for tourists or something like that, plus Te Papa is here so if you want to learn anything about NZ you have to come here. I haven't actually explore the South Island but if you want to explore wild mountain areas that is the place. Glaciers, whale watching, hiking, kayaking etc are all things to do in the South Island. And if you are an adrenaline junky you should visit Queenstown which proclaims itself as the adventure capital of the world. Since NZ has less interesting wildlife than Australia NZ's strength is its natural beauty.
But 2 months is not enough time to do all that so you will have to pick and choose.