As with New Zealand, if you are looking at requesting a UK birth, marriage or death certificate, you will need to pay for this. You can order the certificate you require online at The General Register Office .
In the first instance though you will want to review the Birth, Death and Marriage Indexes to locate the person you are looking for. You can do this through the FreeBMD Index . Pretty much all of the GRO indexes dating from 1837 through to 1970 have been transcribed and searches can also be conducted through commercial sites such as FindMyPast and Ancestry.
The registration of Births, Deaths and Marriages in Australia is the responsibility of the state and territory governments. If looking to undertake a search or order up a document, in the first instance head over to the Government Information and Services portal where you will find links to the various jurisdictions.