Capital of New South Wales. The iconic Sydney Opera House, a hopping night scene, plenty of free parks, and availability of flights make this a great place to start/end a trip down under.

Nav:

Sydney Opera House: This iconic landmark is easy to walk around, though you need tickets (or eat at a restaurant inside) to enter the building.
The nearby Sydney Harbour Bridge offers great views of the Opera House. If you're not afraid of heights, there is an excursion where you can pay to climb the top of the bridge for even better views.

Royal Botanic Gardens: It's a large space and a beautiful (and free) walk that ends at the Sydney Opera House, depending on where you start. The sculptures of the four seasons are beautifully detailed.

ANZAC Memorial / Hyde Park: Also free attractions, the ANZAC Memorial is located in Hyde Park. The memorial started as a tribute to the Great War, World War I. At the time, neither Australia or New Zealand could muster enough troops to create an army corps. They combined forces and contributed the Australia New Zealand Army Corps (ANZAC).

Bondi Beach: A great beach for swimming, surfing, and tide pooling in the city. Accessible by bus.

Sydney Olympic Park: The stadium is currently used for entertainment venues but the surrounding area is still accessible as a tribute to the 2000 Olympic Games.

Nosh:

Scruffy Murphy's: Fish and chips

Mr. ChiXi: I stayed near Chinatown so there was plenty of asian food, which is alright by me. The wonton soup was tasty.

PhoLV: Got the combo pho with brisket, meatballs, and tendon. I don't eat the tendon but allow the collagen to soak into the rest of the ingredients.

Nip:

Sydney Brewery: Imperial Stout. At 9% ABV, you get some bang for the buck.

4 Pines Brewing: This is a local brew as well, located just north of the city but also available at most pubs. The Pacific Ale is worth a pint (or more).