Hi there fellow Expat!
I'm new on the board, but as I've just got myself a referral to a FS. I can help.
You go to your GP and get a referral to the clinic/specialist of your choice. If you are in Adelaide Repromed has an excellent reputation.
It is covered by Medicare and there's no real wait. However, there are gap fees (the balance of the bill have medicare have paid out their maximum benefit) You can look on the repromed website where the out of pocket costs are clearly listed.
I believe that an IVF cycle costs around $2500 out of pocket.... Not cheap but a lot less than paying for it privately in the UK! You will also save yourself a fair amount if you get your notes sent over from the UK so that you don't have to go through all the tests again.
I'm right at the very beginning of my investigations for secondary infertility, so I'm not much help really, but I'm sure that somebody else will be along shortly who knows more than me!
Good Luck!!
Ali x