I always go in their winter, the summer would kill me.
Perth - I have done a trip on a harly here, the wine cruise is fun and the beaches are great. Rotnest Island for the Quokkas and Freemantle to see the old buildings.
Sydney - yes do this, but you can do it in a few days. Go see something at the opera house, you have to walk on Bondi, or learn to surf there.
Then take a Ferry to Manley and explore some history. All the bars are staffed by Europeans - it's the opposite of London.
Back to the Rocks for gift shopping.
Cairns - yes go for it, a trip to the Barrier Reef - with a scuba lesson if you don't already dive. The rainforrest is amazing aand if you stand on the beach where the rainforrest ends you can stand in two world herratige sites at once. I did a trip that took in the Rain forrest, a boat ride to look for crockodiles and a trip to a crockodile farm.
I have not been into the outback - yet, or to Monky Mia to interact with Dolphins and I can't recomend on other cities because I have not been yet.
Check, but I got an 'airpass', basically if you buy internal flights before you leave the UK then you get them cheaper.
How long have you got? Do you want to just be able to say you have been somewhere or do you want to explore it properly? If you have two weeks I would say do a week in Sydney and one in Cairns (or near to).
I have always used public transport (or a relative / friend has driven me).