When I was a kid, I remember grown-ups joking that any deep hole (that I was digging in a sandbox, perhaps) would go "straight down to China." They were wrong, of course, and today you can answer the same question online with Antipodes Map.
The antipode, or opposite point, of pretty much anywhere in Australia is the Atlantic ocean. If you're on the east coast, you'd come up closer to Africa and Europe; on the west, closer to the United States.
Travel the globe a bit, and things get more interesting. Hawaii's antipodes fall in Africa, with Honolulu opposite Namibia and the Big Island opposite Botswana. Meanwhile, the opposite of New Zealand is France. And if you truly wanted to dig a hole to China, you'd need to start in Chile or Argentina.
By the way, if you're serious about digging that hole, I'd recommend the children's book How to Dig a Hole to the Other Side of the World.