My favourite places to visit are:
Melbourne (do a coffee tour and check out the cafe lifestyle in old shopping arcades, go to the Penguin Parade at Philip Island - sounds a bit touristy but if you do the Ranger tour, it's AMAZING!, take 2-3 days to drive the Great Ocean Road, do some wine tasting on the Mornington Peninsula, visit Healesville animal sanctuary, hang out at the beach at St Kilda after some people watching). Melbourne is my all time most favourite Aussie city and I would move there in a heartbeat.
Tasmania - underrated but amazing nature and hiking and the friendliest people ever. Fascinating history, great food and wine, I loved it and will be back soon. So much to see and do but please keep this quiet so no-one ruins it for me. 
Uluru - I HATE tourist traps and I avoided Uluru on many trips for this reason. But I finally made myself go and WOW, just WOW! I was blown away. Somehow the tourists disappear and you find yourself drinking champagne and watching the sun set over Uluru. I did 2.5 days here and it was an experience I will never forget. And not what I expected at all (so much better!)
Sydney - Blue Mountains. Stunningly beautiful and so atmospheric. Get away from the tourists and do some hiking. I got a lift part of the way back and got the boat back into Sydney in the evening. Bliss!! Sydney itself is ok. Nothing to get over-excited about (but I've lived in some very cool cities so I'm probably hard to impress).
Not a fan of Brisbane (meh!) or Perth (the land that fashion forgot) but I liked Fremantle a lot. Heard great things about Kangaroo Island and Coober Pedy so hoping to get there on my next trip.
In terms of stopovers, I prefer to fly straight through from the UK/Europe to Australia, then pop 10mg melatonin at bedtime and I'm on local time the next day. Works every time. The hardest thing about doing the entire UK to Aus trip in one go is that every meal seems to be breakfast (due to direction of travel and timezones). I am mightily sick of breakfast by the time I get there.
I like Qantas for internal flights. Even in economy their local flights are very good and have always been on time in my experience. Their business class Melbourne-Sydney flights are excellent!! And very good coffee in their domestic business lounges. For the long haul, I like Emirates business (on the A380 but NOT the 777) but Singapore Airlines are obviously better (but more pricey). I went BA Club last time and they were pretty solid but having Gold status with them tends to mean I get looked after (whereas status with Emirates never made any difference to how I was treated - either way). Qatar get good reviews and will fund a 48 hour stopover (you contribute $100 only for a 5 star hotel I seem to remember) in Doha. Nice stopover if you fancy having a very cheap Middle East break on the way there/back.