The best price found from United States to Australia is on May 26th, 2015, travelling with Fiji Airways. See more options, compare prices and book: search flights.
High season (with the highest prices) for flights into Australia is roughly over the country's summer (December to February); low season generally tallies with the winter months (June to August), though this is actually peak season in central Australia and the Top End. Australia's international carrier is Qantas, which has an outstanding safety record (...as Dustin Hoffman said in Rainman, 'Qantas never crashed').
Australia has numerous international gateways, with Sydney and Melbourne being the busiest. Other than the big-city airports listed here, some smaller cities (notably the Gold Coast) also offer international flights.