I'm no Aussie, but I do medical transcription for an Australian company (yay internet). Hopefully our Australian friends will correct me but this has been my experience typing out their records.
It is a dual system in which if you can pay for private coverage you can or you can be taken care of in the "government" system. My experience has been that seeing a GP (general practitioner) is not terribly complicated and they are usually accommodating with appointments. It's seeing a specialist that takes time and having surgery too. Those run on categories of urgency. I've seen someone have a heart attack and not get a follow up with a cardiologist for a few months because that specialist was booked so far out (now that doesn't mean the GP may not be doing the necessary tests and intervention in the meantime). I have also seen women with very painful endometriosis go months without their approved surgery because they weren't urgent enough.
If you pay private I believe it's pretty much like here.
So my experience from the other side of the earth is that it's great, unless you need a specialist or aren't in danger of death or loss of limb in the near future.
That's just what I've seen in my work. Hoping our Australian friends can chime in.