If you don’t want to stray too far from the city then you should check out Town Hall and the Queen Victoria Building- catch the train to Town Hall using your trusty Opal Card and a tourist map.

There are tours of Town Hall (see http://www.sydneytownhall.com.au/ for detail) and you are right next to the Queen Victoria Building – a great place to shop or for a relaxing coffee.  From Town Hall you are a short distance from Sydney Tower, Hyde Park and many of Sydney’s large department stores.