They do have fertility specialist in some public hospitals. royal womens has a public IVF program (I am fairly sure) box hill used to have a fertility clinic but they don't do IVF. Where do you live / work - that probably determines where you need to look for FS's? When choosing to go public, balance out the time you may have to wait in a public clinic and think how much that is going to cost you in lost wages. while a private FS will be more expensive, it is easier to schedule time with them.
good luck.