Resolute Beach, West Head, Sydney www.bestsydneywalks.com/resolute-beach-and-resolute-track/ We came across this by accident when we did a bushwalk in the Ku-ring-gai Chase National Park. The walk itself is lovely and you'll see Aboriginal rock carvings, but then you'll find a steep walk down to a beautiful beach with calm water and great views of Sydney Harbour. There is an Aboriginal midden just before you take the track down to the beach. Be prepared for the steep walk back!
North Head Sanctuary, Manly www.harbourtrust.gov.au/visit/north-head-sanctuary-manly This is a wonderful National Trust property at North Head, Manly with old military installations, wonderful tracks to walk or ride bikes through and war memorials. Not to mention amazing views of Sydney Harbour. This is the same area as the Quarantine Station (below).
Quarantine Station, Manly www.quarantinestation.com.au/ The Quarantine Station at Manly is the site of the processing of arrivals to Australia from the 1830s to 1984. The original buildings are there and there are organised tours you can do, but you can also wander around yourself and enjoy the small museum for free. There is also a wonderful little beach with calm waters and a lovely view. You have to park at the entrance and walk down so something to think about. You'll see a photo below of the stairs that take you down to the wharf, but you can also walk around and down a road. There are interesting carvings in the rock face near the wharf which commemorate ships and people who have been to the site - there are 2 photos of those below.
Cockatoo Island www.cockatooisland.gov.au/ If you visit on a Sunday with the $2.50 family ticket on Sydney's public transport, you can take a picnic lunch and explore this amazing harbour island. There are historic buildings to explore, lovely spots with harbour views to have lunch and the children will love the space to run around. The Sydney Biennale is held here as well and that is really worth a visit with lots of art installations the children will enjoy as well, www.biennaleofsydney.com.au/.