The Royal National Park, just south of Sydney, is the oldest gazetted national park in the world, dating back to 1879. Yellowstone in the United States is older but it was originally called a "recreation area", so take that Yanks. Bound to come up in a pub quiz.
There are a number of different areas within the park, however for this time of year you cannot beat Audley which is where Kangaroo Creek meanders into the Hacking River.
There are public BBQs (you can also bring your own portable gas version), pleasant walks, a small cafe, large picnic areas, and you can hire boats and go for a lazy paddle. It's a great place to get away from the city for an afternoon, without having to travel very far.
An official site can be found here but it's not that useful. The Wikipedia page is better. Also see below for a Google maps link.