The best way I've found is to look on www.realestate.com.au or to check the Advertiser on Saturdays for listings of rentals.
I've lived in various parts of Adelaide and found that around Clovelly Park/Oaklands Park area is good, which is near Brighton actually. They're about 20 minutes drive from the city, very close to the Marion Shopping Centre (including Greater Union movies) and new Marion Aquatic Centre, as well as being very close to the beach. Public transport is also decent from here with both train and bus services.
Hi, Send a quick e-mail to firstname.lastname@example.org and say what you are looking for. I came from the UK and the rental procedure is a little different here to back there - the staff at Ray White Adelaide are friendly and will help you with understanding the procedure.