As a recent Adelaide Tourist (visiting our daughter) over a week we did:
Botanical Gardens
Museum and also Immigration Museum
Adelaide Zoo (Panda's cost extra Bux, but good)
Light Rail out to Glenelg Beach.
Port Adelaide (Great indoor second hand market on the weekend, not sure which day)
River Cruise at Port Adelaide is Cheap.
Hire a car and head to Birdwood (National Motor Museum) & Handorf (great model railway display).
Wineries nearby - north of Gawler & Kapunda very close.
We had a great time.