The Secret of Cromwell House by Gerald Stone — a non-fiction account of the Tamam Shud mystery, the most notorious unsolved case in Adelaide's history. Stone investigates the man found dead on Somerton Beach in 1948, whose identity was never established. Essential reading if you visit the grave in West Terrace Cemetery.
Dirt Music by Tim Winton — set in Western Australia rather than South Australia but captures the quality of the Australian coastal landscape and the social world of fishing communities. Winton's closest Adelaide-relevant work is his non-fiction Island Home (2015), about Australian landscape and identity.
The Fortunes of Richard Mahony by Henry Handel Richardson — a trilogy following an Irish immigrant through the gold rush era, set partly in Adelaide during the colony's early years. One of the most serious Australian novels of the 20th century and a window into the world that built the city you are walking through.
My Brilliant Career by Miles Franklin — set in the New South Wales bush but the most important early feminist novel in Australian literature, first published in 1901 by a writer who grew up on the land. For a specifically South Australian voice, try The White Earth by Andrew McGahan, which deals with land, inheritance, and guilt on the Queensland Darling Downs but reflects something common to all of rural Australia.