Welcome to the forum. Lots of information here and a group of people who live with bipolar. Look in the forum Staying Well/This bipolar life. It's a long thread to read all though but I suggest you read the first page at least before going to the end. You can catch up later if you wish. The Black Dog Institute has good information on bipolar. www.blackdoginstitute.org.au There is also a thread in Staying Well/ Useful books and resources
The public hospital system is very good in many ways. The problems arise when you are treated in a clinic where you may see a different psychiatrist every time. They always want your story again so you reach the end of the consultation with not much happening. By the way, do you see a psychiatrist there or a psychologist?
You can see a psychologist through the Medicare system for up to ten visits per calendar year. You need a mental health plane for this. Your GP will write it up and do the paper work. The fees are rebated in the same way as other doctors fees. I do not know what the psychologist charges and neither do I know what the rebate is. It's worth paying a visit to your local Medicare office.
You can get a referral from your GP to see a psychiatrist. This is the same process for seeing any other specialist. Psychiatrists have a large fee I have found and Medicare does not pay a huge amount. However there is a safety net which tallies the gap payments and when you have reached the cut off point your benefit reduces quite considerably. Again have a chat with your local Medicare office.
The remainder of your questions are best answered in the This Bipolar Life thread. There is so much information there which is freely shared with others.