I had longer than you in Oz so I went right up the red centre from Adelaide to Darwin, stopping at Alice Springs, (for Ayers Rock), and then Katherine. But someone I met travelling on the way up the east coast flew from Cairns to Alice Springs (which is 6 hours by bus from Uluru or you can take a half hour flight to Ayers Rock from Alice).
What I'd suggest then (if you dont' mind a suggestion!) is that you could do the same as my friend, fly Cairns to Alice, head to Uluru (most people stay a couple of days, you can either hire a car or take a tour), and then from Alice Springs, fly straight to Melbourne, (my favourite city in Oz) - really vibrant and arty, good for a few days, and you can also do a day tour along the Great Ocean Road from there - I can't remember the company I went with, but it was really good, just a small group and included a helicopter ride over the 12 Apostles which was unmissable. Then from Melbourne you can head to NZ easily (it's about 3 hours flight to Auckland from Melbourne as far as I remember).
If you go on to this site:
Webjet - this can sometimes give you the cheapest fares on domestic flights in Oz - though not always, so check - generally Jetstar and Virgin Blue will do cheap flights and have special offers at certain times, so go on to their websites too. It won't be hugely cheap as you'll be in high season but it shouldn't be too bad if you hunt around. Also Qantas will do multiple route tickets, so with them you could book Cairns-Alice-Melbourne all on the one ticket and that might be a better price.
I think in 6 weeks you could easily do all this - ie a month up the east coast (maybe a few days longer) and then 10-12 days for Ayers Rock, Alice, Melbourne, Great Ocean Road. Ok I'm done with the advice!
My favourite places up the east coast were: Sydney, Byron Bay (good chilled out town, will be busy when you're there, but still nice I think - surfers' and boogie-boarders' town, also an alternative scene there), Fraser Island, Whitsunday Islands (fantastic for snorkelling round GBR).
Cairns was ok, it's worth taking a trip (you can do it there and back in a day or stay overnight) to Cape Tribulation and go on a nature boat ride down the Daintree river - you'll see lots of crocs!)
Hope that helps 😊
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