Adelaide is nice in itself, the Barossa is awesome and if you're into wine you can spend a lot of time there, the Finders Ranges offer beautiful mountain desert experiences. Kangaroo Island is a nature lover's paradise. The beaches are also wonderful, areas like Robe are exceptionally popular, and if you want an adventure to the west I remember Ceduna as being lovely. The Coorong is incredible and birds abound on its wild and windswept sand reaches and wetlands. To the east areas like Mt Gambier and the limestone caves of Princess Margaret are a different experience again. To the north Coober Pedy is another unique experience.
South Australia is vast and varied and I'm so pleased you have lots of time to spend there as there's SO much to see. almost 8 years ago
For short trips from city (30-60 mins driving), the Adelaide Hills are really beautiful and there are some nice walks (also some nice wineries!). Although give Hahndorf a miss unless you fancy a mega dose of Tourist Town.
With more time, I'd suggest heading over to Kangaroo Island for several days. Some of the coastline is very remote and windswept and quite beautiful.
Also the desert regions are great - start in the Flinders Ranges and head North. As people say, there's nothing there, but you've GOT to go and see it. I've spent several nights free camping in the desert regions and loved how isolated it is. Maybe take a friend with you... and fuel and water. But the star gazing makes it worthwhile. almost 8 years ago
All great suggestions to date. If you want a slower pace try Lake Bonney in Barmera. 2 hours west (towards Renmark). The lake and surrounds are great for all water sports or just lazing on the beach by the lake. Go on to Berri, Loxton, Paringa and Renmark. If you like hiking there are lots of trails in and around Loch Luna (near Barmera). When you get to Barmera try the local vanilla slice from Barmera Bakery - it is a real sweet treat. about 7 years ago