There's a museum at the start of the base of the harbor bridge that gets you a great view for a few $ instead of what they want to climb the bridge which is steep ( pun intended).

Also must walk through arboretum near opera house as there are millions of bats ( the size of dogs with wings) hanging around the trees during the day and then at dusk they start flying away.
Manly beach ferry is a great day trip as you get harbor tour via ferry and beach area can feel like Santa Monica on nice days. At end of manly beach is a man made ocean public swimming pool that is pretty neat to go to. Good walk along beach to get there.
Also although a bit of money but worth it is the jet boat rides from circular quay. The price for a ride is negotiable. In fact if its slow I've gotten two for one a few times especially if you tell them you are a repeat customer or a delta employee or us military. They basically look for excuse to cut price. If very busy though with lots of tourists then may not work as well. Great time especially with kids.
I have three kids and we've gone to Sydney three times in last 6 yrs but we live in hawaii so commute is much shorter for us ie 9 hrs.
Aloha luv