Everyone (except New Zealanders) needs a visa, but for most countries an electronic visa, called an ETA, is all that is required. This can easily book this online at
www.immi.gov.au.
I would probably choose Sydney over Bryon Bay for New Years Eve. Sydney is a big city with around 5 million people and they always put on a good show for NYE with a fireworks and lots of parties. In contrast Byron Bay is a small town with a laid back atmosphere. You can have a good time there, but it is not the party party atmosphere that you find in Sydney. The only drawback with Sydney is that you will need to book your accommodation in advance as it is a popular spot for New Years Eve and many hostels jack up their prices from NYE.
The flight sounds expensive but the time between Christmas and New Year is the most expensive time to fly. You may want to look into a round-the-world flight, they are often around the same price as a direct flight and you get lots of stops in exotic places that you may not usually get the opportunity to visit.