If you are looking for an adventure then a Harbour Bridge climb should be on your bucket list.
Take a train to Circular Quay using your Opal Card (or taxi if you prefer) and you will see the Harbour Bridge. You must book in advance if you plan to climb the Bridge or you may be disappointed. Information about can be found at http://www.bridgeclimb.com/
After the climb you take investigate The Rocks, one of the oldest parts of Sydney. There are museums, art galleries, pubs, restaurants and a whole lot more – for more details check out http://www.therocks.com/