As a new Fellow you will need to reapply for a medicare provider number for wherever you are going to work as a GP. You will be Vocationally Registered (VR) which makes the whole process more streamlined than when you were a GP registrar. You should contact Medicare directly. The Department of Human Services provides information on applying for a Medicare provider number.
If you are continuing to work at the same location where you have been a GP registrar, you can notify Medicare and they will simply remove the expiration date from your current Medicare provider number.
You will also have the option to be registered with the Medical Board of Australia (through AHPRA) as a Specialist General Practitioner. This requires a small fee and more paperwork but it will mean that you have both general and specialist registration (please note that while this is not a necessary step, it is quite nice). Here is a useful starting point:
It is also critical that you notify your medical indemnity insurance immediately of your change in status to working as an independent VR GP. You will likely pay an increased premium.
Lastly, if you are planning to work as a sole trader/contractor GP rather than an employee (which is most common), then it would be wise to talk to an accountant about how to structure your finances and prepare for your tax and superannuation.