The public schools are zoned- you have to live in the right area to guarantee a place. It might not be the closest school to your address that you are zoned for.
Many schools do not enforce the zoning as they have sufficient places to accept anyone who wants to enrol. But schools - particularly those in high demand- enforce this.
Schools you mentioned- Adelaide High, Norwood Morialta, Marryatville and Unley are high demand and do enforce zoning. The exception would be if your child excelled in an area of the schools specialty- ie music or a particular sport.
Look at the Department for Education and Child Developments website for a map of the school zones. This will help you to make sure you find a house in the area zoned to the school you want.