There is a first class bus network that links inner city Brisbane with outlying suburbs and the Sunshine Coast to the North and Gold Coast to the south. A train network also runs between the city centre, inner and outer suburbs and both coasts. Train journeys also connect directly to the airport (called the air train) to provide maximum convenience.
Your best option for getting anywhere in Brisbane via public transport is by putting in the to and from address into the Journey Planner on the Queensland Government TransInfo site:
http://www.transinfo.qld.gov.au/