We too are moving to Adelaide in the next month or so having just sold our property here in southern Tas. I hope others keep posting recommendations and advice.
Do be aware that in the northern suburbs of Adelaide, although all the rental prices are much much cheaper do you have safety issues. They are a long way out and due to the cheaper prices have attracted people with issues. I recently had to dissuade somebody from Sydney from considering renting out there. Regrettably Im talking about Elizabeth suburbs and Davoren Park. I found living in the north east of Adelaide, Klemzig and Valley view was generally safe and a good distance from the CBD. Of course if you can afford at the eastern suburbs are generally pretty good. Depends where youre going to work of course. Adelaide is commuting times are not as long as those in the eastern suburbs but they are getting longer as we dont have the freeways those states to do and our public transport is not nearly as good. I always found commuting particularly when I had to do an hour each way pretty tiring. Good luck with it all.
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.
You can search right here on Homely.com.au for places to rent and buy (as well as the only portal that has suburb reviews from the community)
Start by contacting a central office like ray white Adelaide. Ask what properties they have on their books and then Realestate.com.au
realestate.com.au would be your best first step but don't ignore the advertiser on saturday as many agents advertise their rentals in the press as well