For your health insurance options go to 'Health'.
Call either the Ambulance (Tel: 000 [triple zero]) or go to the nearest ED Department. Make sure you know the closest hospitals with 24h ED and have their numbers ready - especially if you have
The first Doctor to see is always the GP (General Practitioner). You need his referral to see specialists and in contrary to Switzerland, even to see an obstetrician/gynaecologist or paediatrician. Thus, if your child has the flu you see the GP. Ask your work mates or neighbours for a GP they can recommend so you have a number ready in case you need one.
The National Immunisation Schedule of Australia slightly differs from the one in Switzerland. If you are immunising your children, discuss the required immunisations with your GP. Immunisations are done by the GP. Councils provide free immunisation services if you are eligible for Medicare (only permanent residents).
Whereas health checks for children are done in Switzerland by the paediatrician, they are done in Australia by a GP or by a nurse in a Child and Family Health Centre. These centres also provide services similar to those of the Swiss 'Muetterberatung' (example: Parenting and Child Health SA, List of Centres).
Having your first baby in Australia, probably far from your own family can be a challenge. Many organisations provide help via phone or online.