Adelaide is a very laid back city, very wide roads and lots of good shopping and eating. The Rundle Mall is a pedestrian shopping area. The Central Markets are worth a visit (open Tues, Thurs, Fri and Sat morn)
lots of stalls where you can try out different foods and some good Cafes.
If you like Museums and Art Galleries there are lots, I think the South
Australian Museum is the one with all the Aboriginal artifacts. I enjoyed the Migration Museum it tells the story of the early migration of the Europeans. They are mostly situated on North Terrace.
Adelaide's most popular beach is Glenelg and you can catch the old tram from Victoria Square, it takes about half an hour.
Hahndorf is about 30km south-east of Adelaide, it's a popular destination
its an authentic? German Village, complete with Antique Clock Museum and a brewery. I think you can go by local bus or a tour.
The Clare and Barossa Valleys are all within reach of the City, really pretty areas where you can visit the Vinyards and taste the wine. I'm pretty sure there are day tours if you don't have a car.
Another Area to visit if you have time is the Flinders Range, not far from Adelaide, a mountainous area with lovely scenery, and Kangaroos, Emus etc.
You can also take riverboat tours along the Murray River.
I spent a couple of days in Adelaide in September last year, I was only staying there as a stop off on my way out to the bush and eventually Alice Springs and didn't know much about the place but was quite taken with the city and people.
I hope you enjoy your ride on the Ghan, we had to wait for it to cross the road at a crossing when we out near Coober Pedy, it seemed to take forever to go by.
Taxis are not expensive and usually the drivers are only to willing to share their knowledge of the city with you.