Belair National Park is a great place to visit any time of year. It's the perfect spot for a family gathering and there are plenty of things to see and do. The park has 54 tennis courts available for hire, BBQ facilities, an adventure playground and you can also hire the main pavilion building.
Tours of Old Government House, which was originally built for the governor of the day to reside in during the summer months, are run every Sunday and all public holidays. Be sure to check with the park information office for tour times and costs.
The park also has several marked trails and tracks allowing you explore the beautiful natural setting at your own pace. Something really worth checking out especially after the city has received a lot of rain is the upper waterfall. Make sure you take your camera as you are very likely to see some of the native fauna living in the bushland.
Belair National Park is a great example of environmental preservation and also has it's own nursery dedicated to several Australian native flora species. Also located within the park are caravan and camping grounds so why not make a weekend of it.
For another memorable Adelaide experience head a short distance out of the city taking the Upper Sturt Road at Belair and follow the signs.