Sydney has more natural beauty than any other major city in the world - at least, any other that I've visited.
The harbor provides miles of coastline and beaches. The Blue Mountains are about 2 hours away by train and offer hiking and small town exploring. Sydney, itself is a culture hound's heaven.
Budget Sydney definitely includes walking, which is great because there are so many wonderful neighborhoods to explore. High on my list are: The Harbour Bridge north to Kirribilli and Blues Point which includes the Secret Garden. The Harbour Bridge south to the Sydney Opera House, The Royal Botanic Gardens and Art Gallery of NSW. The Harbour Bridge south to The Museum of Contemporary Art and The Rocks for shopping (markets Friday, Saturday and Sunday). Neighborhoods such as: Surrey Hill, Chippendale, Glebe, Kirribilli, The Rocks
There are many beaches accessible by ferry and public transit from Sydney including: - Manly Beach; - Balmoral Beach; - Bondi Beach.
Hang out by the pool. The Olympic Pool in North Sydney is in a spectacular setting. There are views of the harbor and the Opera House.
Top of the Shangri-la. Free to enter, buy a drink or not, and enjoy the view of Sydney Harbour. It's especially dazzling at night.
Pack a lunch from a specialty store or market. The beach is a big thing in Sydney as are parks and gardens as noted above. Pack a lunch from one of the many specialty food stores or markets and have a picnic in the fabulous outdoors. Seafood is a must in Sydney. Go to the Sydney Fish Market to see the mongers selling their wares and enjoy a fabulous selection of fresh sushi and other delectables for lunch.