Id suggesting seeing a GP as soon as you get here and they will talk to you about public or private: obstetrician , shared care or midwife care- birthing unit , and then they will arrange a referral for you. If you get organised soon after you arrive you should be able to get into your nearest public hospital or have a good choice of private obstetricians
This is the link to the website for the main "Womens and Childrens hospital" in the centre of the city
http://www.wch.sa.gov.au/services/az/divisions/wab/obstetrics/
There are also several other public hospitals that do obstetrics in North-Lyell-McEwin Hospital; west- The Queen Elizabeth , and south - Flinders Medical Centre, amoung others. Services are fairly similar. Many good private hospitals but even with private insurance still fairly large bills for you to pay on top. Ive had children in both private and public hospitals here and both were good, Im sure you will get things sorted quickly and easily.
Good ante-natal aqua-aerobics and exercises classes etc, some held in gyms with childcare ; and wonderful mothers groups for afterwards- great way to meet some new people when you get here