There is only one airport serving Sydney, this is located not far from the city centre, it is only 8 kilometers that separates Downtown from the airport. There are many ways to get to the city, it may be the train, private shuttle, taxis or there are passengers that even use a bicycle. This is the busiest airport in Australia and the main hub for Qantas the flag carrier of the country.
It is good to keep in mind that there are three terminals, the international terminal is completely separated from the other terminals and as such there is a need of longer -at least one hour and 15 minutes- when one is connecting from international flights to domestic or vice versa. There are free shuttles running between the three terminals.
Train is probably the best way to reach the city. There are two terminals to board the train, one is the domestic that is located between terminal 2 and terminal 3 and can be reached from the arrivals hall. The other terminal is located at the Northern end of the international terminal and it is as well accessible from the arrivals hall. The whole trip to the city takes less than 15 minutes and there are trains every ten minutes from 5:00 am to 2:20 am during weekdays during weekends the last train runs at 1:00 am. The price for an adult from the domestic terminal is 16. 40 AUD this is around 14. 50 USD, from the international terminal it costs one dollar more. Children (younger than 16) cost 12. 50 AUD and families only have to pay for one child. Children under four travel for free. This train connects with the whole Sydney metropolitan train network so it is very easy to reach almost any point in the city.
There are several shuttle buses that take passengers directly to their hotels but this option has to be precooked online. It is possible to reach the city by public bus, the line number 400 leaves both terminals around every 20 minutes and it cost from 2.60 to 4.70 AUD depending the final destination, there are several available passes but one can buy single rides at the main terminals or onboard, for more information you can visit the main webpage of Sydney buses http://www.sydneybuses.info
There is a bike path that links the international terminal with Mascot and from there it is is possible to get to the city centre. There are bike racks available at all the terminals.
It is possible to catch a taxi, there are marked spots where one can board official cabs, prices vary according to the destination but it it approximately 55 AUD to the city centre, passengers have to pay for any toll on top of the fare.