Very close to the city of Adelaide, on the eastern side, at the base of the Adelaide foothills, is a beautiful little spot. This is Morialta Conservation Park. He you can spend the day, having a picnic at one of the charming little spots.
A walk to the First Falls is a must. The walking track meanders its way up to the falls, passing over Fourth Creek quite a few times. On the way to the falls, you can climb up to the Giant Caves... I remember going there as a school girl, my whole class fitting into the cave.
The falls are spectacular after a heavy rain. Thats the best time to go.
For the more adventurous, there is rock climbing. This is done of Norton Summit Road.
Do be careful, as many people have had to be rescued.