Riding the public transportation is Sydney is super easy. We found it to be the most economical way to get around. You can buy an Opal card at train station at the airport and load it with as much as you want. Use your Google maps to figure out how to get to your destination and top up your Opal card at any of the train stations and many grocery stores. The Opal card is good for the ferry as well (not the Fast Ferry).
All the details you need for the Opal Card can be found here: