Clearly, Australia is a huge country, this means if you want to see it all, you are going to have to do a lot of travelling, you've got a few options but time must be allowed for your movements from place to place.
For a stress free tour of the east coast, we highly recommend using Oz Experience, you can choose from a number of different routes before you leave and use their hop-on, hop-off service to visit all the best spots. It's great fun and you'll make loads of friends, the drivers are very knowledgable and will give you all the information you need you need along the way.
For a more individual trip, a great option is the Greyhound bus service, this is a national company and can get you almost anywhere, with cheap rates this is a great option, a lot of the services run overnight so you'll save money on accomodation as well.
For the ultimate trip up the East Coast, grab a camper van, most companies will rent vans sleeping 2-6 people and allow you to pick up the van in one place and let you drop it off in another. It's an awesome choice for those of you wanting the freedom to do whatever you want, whenever you want.
Internal flights are also a great option, there are plent of airports so these are a good option to save time travelling around the vastness of Australia