From Sydney Airport to to the city centre:
When you arrive in Sydney, the easiest way to get to the hotel is by taxi (20-30min). A taxi costs 50 AU$ + 4.25 AU$ for tolls.
If you are 1-2 people and your hotel is next to one of the following train stations (Central, Museum, St James, Circular Quay, Wynyard and Town Hall), then you better take the Airport Link (15min - 17,90AU$). See a Sydney map with all stations marked.
Don't forget to get a city map from the airport. The best way to get to know city is roaming its streets, so you'll need a map to avoid getting lost!