There is no prize for doing this quickly. Almost nothing here has a deadline in the first week, and several of the things people rush are the things that cause the most trouble later.
This is the order that tends to work.
First, and only this
Register the death and order certified copies of the death certificate. The funeral director usually arranges the first batch as part of their service.
Order five to ten. Nearly every institution wants an original or a certified copy, and most of them will not give it back. People routinely order two, then spend the following month ordering more.
Then, one form that covers many
The Australian Death Notification Service is a single online form that notifies many banks, telcos and utilities at once.
It is free, it is run by government, and it is the single biggest time-saver available. Do this before you start ringing individual organisations, because it may have already handled several of them.
Then Services Australia
One notification to Services Australia covers Centrelink, Medicare and Child Support together. A bereavement payment may apply.
Then the post
Australia Post mail redirection or a hold is cheap and immediate. It matters more than it sounds: an unattended letterbox at a house everyone knows is empty is a genuine fraud risk, and the mail itself is often the only remaining trace of a policy nobody knew about.
Five things not to do yet
Don't cancel the mobile number. It is the two-factor key to email, banking and nearly everything else. Redirect it, keep it live, and cancel it last. This is the single most common expensive mistake.
Don't rush to close joint accounts. Check what is still going in and out first — a mortgage, a utility, an insurance premium that keeps the house covered.
Don't distribute anything. Not money, not jewellery, not the car — not before the executor has authority. Well-meant early gifts are the most common cause of family disputes over estates.
Don't pay the deceased's debts from your own money. They belong to the estate, not to you. Paying them personally can be very hard to unwind.
Don't throw out paperwork for at least a year, even when it looks like junk mail. An annual statement from a fund nobody knew about is exactly what an unclaimed insurance payout looks like from the outside.
The part that takes the longest
Not the notifications. The finding.
Working out which banks, which super funds, whether there was insurance, where the will is, who the solicitor was — that is where the months go. Every institution needs to be identified before it can be contacted, and the person who knew is the person who died.
If you are reading this because you are in the middle of it, the honest advice is this: write yours down before someone has to do this for you.