As a general advice: Documents many times needed in these first days or weeks are your passport, the job contract (if applicable), your visa grant letter and a proof of a residential address. Thus have them handy or carry them along while sorting out your business!
Also, do not forget to register with the Consulate General in Sydney! This has to be done within 90 days after the announcement of departure to Australia. The Consulate General is your point of contact for all administrative matters and when you find yourself in difficulty. Registration is not only mandatory but you do yourself a favor as it will speed up processes like getting a new passport.
Most people find a home to rent within 1-3 weeks. Have a look from home on realestate.com.au.
Bringing any important documents you have (e.g. proof of an income). Some banks will allow you to open a bank account whilst still in Switzerland (e.g. try Westpac).
Most people here have and need a car! Once you know which car you want to buy, it will be a matter of half an hour and you drive off in your own car, insurance and registration done by the car dealer. Have a look from home on www.carsales.com.au
A Tax File Number (TFN) is a unique number issued by the Australian Government. You most likely need a TFN for your job to avoid tax withheld from your pay or from the earnings on your bank account. You can apply for it online.
Medicare is the name of the Australian public health care system. Swiss Immigrants with a Temporary Visa are not eligible to claim benefits from Medicare, only permanent residents. Make sure that you have adequate health insurance.
Formal schooling in Australia starts with a kindergarten or a preparatory year followed by 12 years of primary and secondary school. If you know in which area you will be living and to which school you might send your children, enrol them from back home. Public Schools are usually zoned, that means children living in the school zone have priority in getting a position. Private Schools often have a waiting list. A fee applies in general for the enrollment.
CSA Oceania, the council of the Swiss abroad delegates for the Oceania region have a website! Created in October 2016 to improve information exchange and be present as delegates online.