I think you will love Adelaide!
Did a little traveling at the start of the year to Sydney, Brisbane, and the Gold Coast. Even all the extra stuff there was to do for someone my age in those cities, i could not spend maybe more than a week or 2 there at a time. People are friendlier, its not so rush rush, and i believe just a better place to raise a family.
If youre working up Greenhill Rd, id look at areas around Maylands, Firle, St Morris. These are <5mins to the Norwood Parade (great strip for foodies and to take the kids), and more affordable than the aforementioned suburbs like Kensington/Burnside (still only a few minutes from these also). I think you may be able to get into the Norwood Morialta School Zone, or Marryatville, which are up there with the best public schools in Adelaide.
Try staying away from areas like Prospect id say, purely because you would have to cross through the city to get to work, and that will add a fair chunk to your daily commute.
In the areas i mentioned, youre 10mins from the city, 25 minutes to the beach and airport, and an easy drive to the foothills.