Australia is a big place and driving can literally be a killer.
It seems like you only want to do a relatively small area of oz and with 3 months up your sleave you won’t need to rush.
As mentioned the greyhound buses and other companies are good and pretty versatile but they only drop you off at main towns so you won’t be able to do much back beach exploration.
Motorbike sounds like fun but could be a hassle in regards to the distances and the amount of gear you can carry.
Getting your own cheap car is a good option but be aware that insurrance can cost a bit.
There is a big backpacker scene in Australia and some of these tourists know more about getting around on the cheap than the locals do. I know in Sydney they have a backpacker car market at Kings Cross in the city. Backpackers buy cars there, drive them around the country then bring them back and sell them. You might even meet up with another traveller who has a similar trip planned and you can go halves. Some cars are sold with camping gear as well.
Just to give you an idea of distances it takes 10 hours driving just cruising to get from Sydney to Byron bay. But there is a lot of surfing to do in between so why rush.
They may have a similar backpacker set up in Brisbane which might be easier in regards to driving.
To get you in the mood try and check out the dvd “morning of the earth”. It’s old school but good stuff.
Or for something newer I like the feel of “shelter”
Here is a joke about Aussies as well. Might help understand our mentality.
Four men were sitting around a conference room table being interviewed for
a job.
The interviewer asked, “What is the fastest thing you know of?”
The first man replied, "A thought. It pops into your head, there’s no forewarning that it’s on the way; it’s just there.
A thought is the fastest thing I know of."
“That’s very good,” replied the interviewer.
“And now you, sir,” he asked the second man.
“Hmmm, let me see… a blink!,” said the second man. "It comes and goes and you don’t know it ever happened.
A blink is the fastest thing I know of."
“Excellent!”, said the interviewer. “The blink of an eye. That’s a very popular cliché for speed.” He then turned to the third man who was contemplating his reply.
“Well, out on my dad’s farm, you step out of the house and on the wall there is a light switch. When you flip that switch, way across the field the light at the barn comes on in an instant. Turning on a light is the fastest thing I can think of.”
The interviewer was very impressed with the third answer and thought he had found his man.
“It’s hard to beat the speed of light.”, he said.
Turning to the fourth man, an Australian, he posed the same question.
“After hearing the three previous answers, it’s obvious to me the fastest thing known is diarrhoea,” said the Aussie.
“What!” said the interviewer, stunned by the response.
“Oh, I can explain,” said the Aussie, “You see, the other day I wasn’t feeling so well and ran for the bathroom. But, before I could, think, blink,or turn on the light, I shit my pants.”
He got the job…