Sydney's public transport network now runs on a smartcard system called Opal (www.opal.com.au).
The card can be obtained (for free) and loaded with credit (minimum $10) at numerous newsagencies and convenience stores across Sydney. When commencing a journey you’ll need to touch the card to an electronic reader, which are located at the train station gates, near the doors of buses and light rail carriages, and at the ferry wharves. You then need to touch a reader when you complete your journey so that the system can deduct the correct fare. You get a discount when transferring between services, after a certain number of journeys in the week, and daily charges are capped at $15 ($2.50 on Sundays). You can use the Opal card at the airport train stations, but none of the aforementioned bonuses apply.
You can still buy single tickets (Opal single trip tickets) from machines at train stations, ferry wharves and light rail stops, or from the bus driver. These are more expensive than the same fare using the Opal card, so there's not much point unless you don't think you'll use $10 worth during your Sydney stay.
You can purchase a child/youth Opal card for those aged four to 15 years; they travel half-price. For student and pensioner discount Opal cards, you have to apply online.
Water taxis are a fast way to shunt around the harbour (Circular Quay to Watsons Bay in as little as 15 minutes). Companies will quote on any pick-up point within the harbour and the river, including private jetties, islands and other boats. All have a quote generator on their websites. It's much better value for groups than couples.