Getting the birth certificate should be the easy one. You contact the office of births and deaths in the territory where you were born:
Weirdly, the best list of Australian agencies I can find is in New Zealand... go figure...
https://newzealand.embassy.gov.au/wltn/birth_certificates.html
So, just picking one at random, if you were born in NSW:
www.nsw.gov.au/births-deaths-marriages
Following the links through there...
https://www.nsw.gov.au/family-and-relationships/births/birth-certificates#proof
You must provide a least 3 forms of current identity, one of each from Categories 1, 2 and 3.
If you cannot provide proof of identity from Categories 1 and 2, you must still provide at least 3 forms of identity. At least two of these must be from Category 3.
If cannot meet these requirements, please contact us for further advice.
Category 1
If born in Australia:
An Australian birth certificate
Record of immigration status:
Citizenship certificate
New Zealand citizenship certificate together with passport
New Zealand birth certificate
Category 2
Australia's driver's licence
Australian passport
Firearms licence
Foreign passport
Proof of age card
Category 3
Medicare card
Centrelink or Department of Veterans Affairs card
Security/Crowd control licence
Tertiary education institution ID card
Category 4
Recent utility account with current residential address (issued within last 3 months)