Went once, loved it: went something like this.
Sydney: jetlagged to fuck. Stayed in hostel with awful 22 year olds (I was 23), got no sleep. Cried. My friend was SO TIGHT. Conversations went like this: "shall we stop for a quick coffee?", "Oooohhhh... Reallllllyyy?"
After that really found my groove and had a much more positive experience
Brisbane: nice city, beautiful city
Cairns: LOVED. Good vibe, tropical and quite strange physically but the people are cool
Alice Springs/Uluru: I wouldn't go to Australia without doing this part. Something truly otherworldly about it all. Very VERY interesting and really quite imperative to see
Darwin: eh. The national parks around it are very cool for aboriginal history but the city itself is just there
Adelaide: LOVELY
Melbourne: My favourite. Feels like a southern hemisphere London more than Sydney did (as a lazy comparison Melbourne is more European and Sydney is more American)
Didn't see any of Western Australia but I've only heard good things.
Also (sorry) I MASSIVELY preferred New Zealand just in case you're able to see there also. The most physically beautiful place on earth.