You have 2 options. Firstly, if you have lots of luggage, or heavy luggage take a cab, otherwise the train is a good option.
To get to the train station on exiting Australian Customs turn right and follow the train signs all the way to the end of the terminal and take a number of escalators down to the train station.
Ask at the ticket office for 2 tickets. Firstly an airportlink return pass (approx. $18) and secondly a Mymulti zone one weekly pass (approx. $41) that lasts for 7 days. Even if you are staying for 3 or 4 days, the Mymulti is worth it as you can use it on an unlimited basis on trains, ferries and buses in all the zone 1 areas, including the Manly Ferry. Buying tickets each time you want to use public transport is expensive and inconvenient.
Please note: Use your airport pass to get through the gates at the airport to get to the platform. The next time you pass through station gates to exit Kings Cross use your Mymulti. You do not use your airport pass again until you exit the gates at the airport to go to the airport terminal. To summarise, you only use your airport pass twice, once to enter the airport train platform and once to enter the international terminal. We found this very confusing for a while after much trial and error!
Click on the link below on how to get to Kings Cross Station
Click on the link below on how to get to International/Domestic Airports
My Multi Zone One weekly pass
As I explained above, this pass is really worth getting. It lasts for 7 days and enables you to use trains, buses and ferries on an unlimited basis. Zone One gets you on all the inner city train lines and many close suburbs, city buses as far as Manly and Bondi and all the city Ferries, including the Manly ferry, but not the Manly fast ferry.
You can see so much in Sydney by walking. Buses are really convenient for those short hop on or hop off rides and then let your legs do the rest!