This is an awesome coastal walk - absolutely must do.
The Ferry to Manly is a beautiful 20 minute ferry ride across the Harbour. It leaves from Circular Quay every 20 or so minutes and takes you to Manly (an northern beach town). From there you can do numerous coastal walks, grab lunch or drinks anywhere nearby and chill on the beach.
Iconic bridge climb. Requires booking in advance and you definitely don't want to have a fear of heights!
The main part of the Harbour where the Opera house is. You can grab a coffee or a bite to eat and watch the massive cruise ships go by.
Great for a day where you just want to walk around, be in the sun and hang with some beautiful plants. The birds in the gardens are also amazing, and it's right by the Opera House so you can make a whole day of it!
Clifftop harbour walk with views of the ocean and Sydney harbour. Grab lunch in Watson's bay after your walk! Doyle's fish and chips are great (right on the beach and next to the boats).
Amazing art gallery if it's a rainy day!
Sydney's main zoo with many animals and harbour views. You can take a ferry to the Zoo from Circular Quay.
Royal National Park is about a 45 minute drive south.
They're called the Blue Mountains because the type of Eucalyptus trees there sometimes make the air look blueish :) You can take the train to the Blue Mountains. But you may prefer to hire a car to drive yourself.
Symbio is a wildlife park that has all the classic Australian animals. It is a bit farther to get to, so you'll want to hire a car for this.