Sydney is famous for its bridge. Check out the bridge close up by walking across it! The views are fantastic, the photo opportunities abound and it's free. Easiest way is to catch the train across the bridge from Wynyard to Milsons Point (just one stop, takes 2mins). At the station exit, you should see signs directing you to the stairs you climb to walk across the bridge. One side is for pedestrians, the other side for bikes. The walk takes only half an hour at an easy pace. When you get to the other side, follow the signs to the Rocks, an historical area of Sydney, which is another "must-see" for visitors. Cobbled streets, authentic souvenirs, great restaurants in this district.
If you want to see the Opera House close up, and you can't afford tickets, or everything's booked out, try for a tour (booked through the House itself) or just wander around. Quite a lot is open to the public and you may enjoy coffee or a drink while sitting outside, enjoying the view or the atmosphere.
Both the Bridge and the Opera House enjoy harbour positions. Take some time to wander around the harbour foreshore of Circular Quay. Terrific ambience on any sunny day, especially weekends. Sydney Harbour is beautiful to look at, but even more fun to ride on. No visit to Sydney is truly complete without a harbour cruise of some sort. There are classy evening cruises, with sumptuous food, and basic cruises, operated by Sydney Ferries, that cost very little and probably offer an informative recorded guide as well. The easiest option, however, is to take an ordinary ferry. These run all day, every day, and go to various destinations. Head to Manly (30mins). When you alight, walk straight ahead onto Manly Corso and enjoy the buskers, the ice-cream shops, cafes and restaurants. Stroll the length of the Corso and you'll find yourself at Manly beach. You'll easily understand why it's world-famous.
If you have more time, check out Darling Harbour, Taronga Zoo (accessible by ferry), the Powerhouse Museum, tour the Olympic Park site with its world-class sporting venues (at Homebush) or venture further into the beautiful Blue Mountains. Just a few hours south-west by car will take you to the nation's capital, Canberra, which has a lot more to offer, including the National Gallery and both Old and New Parliament House. Or go directly south to beautiful Wollongong, with its rugged coastline and uncrowded beaches.
Want to touch koalas, kangaroos and other Australian marsupials? Then head to Koala Park Sanctuary at West Pennant Hills in Sydney's Northern suburbs. A well known attraction to locals but not so well known to foreign visitors, here you can feed and hold all manner of Australian wildlife. You can get into the enclosures with kangaroos and wallabies for the real contact experience. Can be done as a day trip with Sydney Day Tours.
