There is a wealth of museums in New South Wales , many (although not all) of which are located in Sydney .  Some of the ones to try and see during a stay in the area include:

• Art museums
• Art Gallery of New South Wales ( http://www.tripadvisor.com/Attraction...) offers exhibits of historical local art, including aboriginal art.
• National Gallery of Australia (NGA),  Canberra
• Manly Beach Art Museum : http://www.tripadvisor.com/Attraction... gives visitors exposure to local art.
• Museum of Contemporary Art  ( MCA): http://www.tripadvisor.com/Attraction...  is the best resource for learning about the art being made in the area today.
• Attractions
• Hyde Park Barracks, Sydney
• St Mary's Cathedral
• St Andrew's Cathedral
• Parliament House, Canberra
• Telstra Tower, Black Mtn, Canberra
• History museums
• Hyde Park Barracks Museum ( http://www.tripadvisor.com/Attraction...) is a unique museum detailing the history of the home where orphans and homeless women once lived.
• La Perouse Museum ( http://www.tripadvisor.com/Attraction...)
• Museum of Sydney : http://www.tripadvisor.com/Attraction...
• Sydney Jewish Museum: http://www.tripadvisor.com/Attraction... details an interesting segment of the history of New South Wales .
• Museum of Australia, Canberra
• Historic Homes
• Elizabeth Bay House: http://www.tripadvisor.com/Attraction...
• Old Parliament House, Canberra
• Vaucluse House: http://www.tripadvisor.com/Attraction...
•  Sydney Town Hall
•  Library
• National Library Canberra 
• State Library, Sydney 
• Natural history museums:
• Australian Museum ( http://www.tripadvisor.com/Attraction...) is known as the area’s first museum.
• War Memorials
• Australian War Memorial, Canberra
• Anzac Parade, Canberra
• Cenotaph Martin Place Sydney
• ANZAC Memorial,  Hyde Park
• Maritime  
•  ANMM, Sydney 
• Science museums
• Powerhouse museum: http://www.tripadvisor.com/Attraction...
• Sydney Observatory: http://www.tripadvisor.com/Attraction... which sits on Observatory Hill offers a view of the city.
• Sydney Tower (and OzTrek) gives the best view of the city and also includes a planetarium; see http://www.tripadvisor.com/Attraction... for information.
•  Sport
• Bradman Museum,  Bowral 
• SCG - Sydney Cricket Ground, Moore Park
• Aussie Stadium, Moore Park
• Olympic Stadium, Homebush
• Randwick Racecourse