Constance Stone: first women registered as a doctor in Australia
Studied USA, Canada & UK. Returned to Victoria & was registered 1890. Established Queen Victoria Hospital in Melbourne, operated “by women, for women”.

Australian Dictionary of Biography (ADB)
Wikipedia
East Melbourne Historical Society: the life of Constance Stone (PDF)
Australian Women’s Register (AWR)
Index of significant Tasmanian women
Encyclopedia of Australian Science

Dagmar Berne: first women to study medicine in Australia.
Accepted at Sydney University 1885. Completed studies in UK and practised there before returning to Australia. Registered in Australia 1895.
(Image from Wikipedia Commons.)

Wikipedia
Dictionary of Sydney
Magazine article, University of Sydney (PDF)
Encyclopedia of Australian Science
Australian Women’s Register (AWR)

Other states

Queensland: Lillian Cooper, 1891 ADB: AWR : Courier Mail
South Australia: Laura Fowler, first female graduate in medicine ADB : Wikipedia : Adelaide Pioneers
Western Australia: Roberta Jull, 1896 ABD : Wikipedia