The Sydney train service is excellent by global standards, although you will hear many Sydneysiders complain, the trains generally run on time and get you within close proximity of all major city and suburban districts. Below is an image of the City Rail Network Map:
From the Airport
All the city circle and stations between Redfern to North Sydney and Bondi Junction cover the city centre. If you are opting to catch the train from the International and Domestic Airport. It is fairly uncomplicated but keep in mind that if you are two travelers heading to the city centre, a single ride (one direction) ticket will cost you $11.80 each. The airport is approximately 15km from the heart of the city so either way you are about 20mins away from your Sydney city.
A Return ticket allows you to travel back and forth between your starting destination and a chosen location. Keep in mind that you cannot use this ticket all day to get on and off. This ticket will cost approximately $6.40 for the entire trip, however drops if you purchase in off peak hours (after 9am)
Also take a look at the MyMultiPasses (link), these offer travel on various modes of transport including Sydney Buses, Ferries and Light Rail.