Ayers Rock/Uluru is probably a must. I'd love to see the outback personally. Perhaps take the train across the continent? How long are you away for?
I haven't ever been to Australia, but I'm in the minority of my friends who haven't. Everyone who has been to Melbourne sings its praises. You could go see the Ned Kelly museum.
My girlfriend quite enjoyed going on a trip to the Twelve Apostles, in Victoria.
And the Gold Coast would be pretty awesome. The beaches look fantastic, though I have a feeling that would be pricey.
AND COME TO NEW ZEALAND! I could show you around Wellington, which was voted Lonely Planet's 4th greatest city to visit (or w/e they were trying to gauge) in a recent poll. It would be interesting to meet you.
And if you brought your tramping gear, you could go on trips that would blow your mind.
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