Below is the link to the PDF file needed to apply for a medicare card. Form 3101.

Print off form, complete and post.

Returning your form

Check that you have answered all the questions you need to answer
and that you have signed and dated this form.
Return your completed form and original or certified documents
to your nearest Medicare Service Centre. For initial enrolments, all
people 15 years of age and over on the application must go with you
to a Medicare Service Centre.

If you live in an area remote from a Medicare Service Centre, or there
is a reason for not being able to attend, you can send your application
together with certified copies of documents and the reason for not
being able to attend in person, to:
Department of Human Services
GPO Box 9822
in your capital city 

Documents required
•     completed statutory declaration 
•     passports for all people listed on the application
•     any 2 residency documents from the list below.  (To be safe more is always better)

Residency documents 
Documents from another country
•     sale of property (sale agreement)
•     cessation of lease agreement for rental property
•     termination of employment (acceptance of resignation by 
•     transit document for household goods and or furniture
•     closure of bank accounts, and/or
•     cancellation of health, property or contents insurance.
Documents from Australia
•     purchase of property agreement and gas or electricity accounts 
in the same name
•     lease agreement for rental of property and gas or electricity 
accounts in same name
•     evidence of employment
•     evidence of children at school or university, and/or
•     private health insurance in Australia, opening of bank accounts, 
property or contents insurance.