Your day trips are great and I've already tried someof them.

A good one people can do close to the city centre is the board walk at Luna Park. It goes to Kirribilli at one end. If you make your way to Kirribilli Ave and walk right down to the end, you can see the Prime Ministers house. At the end of the path there is a little park with a great look out across the harbour.

At the other end the path will take you to McMahon's Point. You'll need to climb up some steps, turn left and just follow the street to a big open space with the best view of the harbour bride and city.