Neither Europe nor North America, naturally, are anywhere near opposite
China. Most of North America's opposite is nothing but the vast Indian Ocean and a few scattered islands; some of the northernmost bits do oppose Antarctica; and digging from Hawaii (though whether that can be considered part of North America is questionable) will get you to Botswana. A small number of people living around Medicine Hat, Alberta might be able to dig to the Kerguelen Islands of the French Southern and Antarctic Lands (a bonafide exotic, remote location if there ever was one!). If you are looking for a dry place to start digging, try Argentina; almost all of its land mass is opposite China, though going beneath Buenos Aires itself will lead you straight to the Yellow Sea.
Australia's opposite number is a similarly boring stretch of the Atlantic, although if you took a boat from Perth a few miles out, you can dig to Bermuda. Africans could only dig their way into the Pacific, except for those opposite the various Pacific Islands.
Europeans and West Asians end up in the Pacific or Antarctic Oceans, except for some Spaniards or Portuguese who could strike New Zealand, a very few French who could hit the Chatham Islands and a few "lucky" Russians in Siberia, which touches the far south of South America and the Antarctic Peninsula in some places. On the other hand, most Southeast Asians do end up in dry land, though it happens to be the Amazonian rainforest. Digging beneath Singapore, for instance, will lead you to an isolated part of the Ecuadorian Amazons, while digging beneath Manila will lead you to a remote part of Brazil's Mato Grosso state, which is less bleak but still devoid of people. In short, all the major cities will not lead you to places of their ilk. Amusingly, though, if you dug straight through from Taiwan (formerly called "Formosa" by the Portuguese), you would end up in the province of Formosa in Paraguay.
70% of the Earth's surface is water. Mostly, that's where you end up. There are a number of Google Earth-based toys demonstrating this, including "If the Earth Were a Sandwich".
To calculate one's antipode, just change the orientation of the latitude (e.g. 44.3 N becomes 44.3 S) and change the orientation of the longitude and subtract it from 180 (e.g. 93 E becomes 87 W). To get a rough idea of it, just use this map
◊ or one like it
◊ (or you can just instantly figure out your exact antipode here).
Of course, you could always dig the hole at an angle instead of straight down. Then you could end up anywhere! But then, that would be cheating, and serious China-diggers don't cheat.
Of utter trivial note: If you dug a tunnel through the Earth (and didn't get all melted or crushed in the process), and jumped in, it would take 42 minutes to come out the other side if you disregard friction (from air, rails, whatever). This is true whether you dig straight through or on an angle. Without some sort of jet engine, you're also unlikely to actually reach the other side, since gravity and inertia work together to ensure that your deceleration from the centre of the Earth upwards will be the inverse of your acceleration during descent towards it, so you'll only travel the same distance in both directions.note In other words, if you jump into an insulated vacuum tube from, say, Hong Kong you will end up at sea level in the middle of the Andes, with several thousand metres to go before you see daylight. Even if both antipodes are at the same relative height, you'd better have someone prepared to grab you quickly at the other end before you start falling back towards your starting point.
Now where's that drill?
