If you're going to do a lot of travelling, get a 10-trip pass. Just be warned - the price changes depending on zones. It can be a little confusing at first. Ask the driver how many zones you will be going or how many times you will need to punch the card for your trip. If you are going to stay in the immediate CBD area, you'll be in zone 1, but zone 2 comes up quickly!
You can buy cards that will cover more zones if you're going to go out of the CBD a lot.
A good way to get to know the layout of the city is to take the Sydney explorer. There are two routes, one which covers the city and the other goes out to the beaches. Here's a view of the entrance to Sydney Harbour - one of the stops on the beaches bus.
The best way to get around the city areas of Sydney is by foot! but if you have a longer way to go, take a bus. Sometimes it can be a little hard working out which bus to take and which route to go on. For more information check out the excellent 'transport info' website (address below) or call 131 500. If you need to travel a greater distance, or you just need to travel the main city areas, a good alternative is to take the train.
Plane is the only way! There is a very handy train from the Airport (A$9) right into the city or there are buses (the yellow ones) to take you into the city for $A9 and there are a number of specialised buses to take you direct to hotels.
Walking will take you far in the city but the much maligned train system is efficient and mostly reliable. These will take you all around the Sydney and surrounds. you can buy day tripper tickets.
Also buses in the city and inner suburbs eg Glebe, Balmain and parts of the North Shore.
The Airport Express bus, operated by State Transit, costs $7 single, $12 return (within 2 months). It allows luggage. It stops at many stops including the railway station, George Street, Circular Quay, then Kings Cross. The travelpass is not valid on this bus.