There is an informal and anecdotal belief among Americans that if they dig a hole deep enough, eventually they would come out in China. This is theoretically possible if the hole is angled in the right way, but if they dig straight down and through the exact centre of the earth from anywhere in the 48 contiguous United States, they'd actually come out in the Indian Ocean. Only in parts of Argentina or Chile would a straight hole emerge in China. Drilling a hole through the centre of the earth from anywhere in Australia would land you in the middle of the North Atlantic Ocean.
The map on this page allows you to approximately locate which place is directly on the other side of the world from anywhere. The complementary red and black outlines are reversed, so that a place in the right place on the black outline map is directly opposite the place on the red outlines. The red outline map is "upside down", with south at the top, so it may be a little confusing to locate places on it.
For example, you can look in the Indian Ocean area of the black outline map, and you can see there the USA in red, upside down between the black outlines of Africa and Australia. If your geography is adequate, you can tell that the furthest away place in the world from Perth, WA, is a spot to the south-east of New York, way out in the middle of the ocean in the vicinity of Bermuda. The closest land on the other side of the world from the Australian east coast is the Brazilian island of Funchal, and its capital, Madeira, which could be reached by digging straight through the centre of the earth from a point some 200 km to the east of Cape Howe on the NSW/Vic border.