Put very simply, Joey, Australia's political system is a little bit like yours. Our States were all separate colonies that decided to federate in 1901. States still have their own governments with responsibility for things like schools, hospitals, law and order etc. The Australian government has responsibility for things like foreign affairs, trade, defence, immigration. State laws do not need to be uniform but do need to be consistent with any Commonwealth (ie Australian Government) laws.
Each state has a Premier, who is elected as head of Government.
The head of the Australian Government is the Prime Minister.
We are a constitutional monarchy. This means our constitution - which, like yours, is written (Britain has no written constitution) - determines that the Governor-General is our Head of State. The G-G is the Queens representative but, contra tarry to popular belief, the Queen is not actually our Head of State. All the powers under our constitution rest with the G-G and the Queen cannot exercise them even if she wanted to. As an aside, this actually means that the debate about Australia needing an Australian as head of state is actually based on a misunderstanding or misinformation perpetrated by politicians looking to distract people from bad policy.
Your Head or State and head of Government are one in the same; the President.
People say that the great thing about your constitution is that anyone can aspire to be president - well, if they're rich enough, connected enough etc.
The real strength of our system is that no one can aspire to be Head of State. Our G-G is appointed based on their suitability for the job, not popularity and therefore they have no promises to keep and no constituency to satisfy. They just need to do a good and impartial job of upholding our constitution. The last thing we need in that role is another bloody politician. The system works great and it would be a shame to *** it up just to get ride of a Queen who has no powers here anyway.