Neither Europe nor North America, naturally, are anywhere [[http://en.wikipedia.org/wiki/Antipodes near opposite]] China. Most of North America's opposite nothing but the vast Indian Ocean and a few scattered islands; some of the northernmost bits do oppose Antarctica; and digging from Hawaii will get you to Botswana. If you are looking for a dry place to start digging, try Argentina; almost all of its land mass is opposite China.
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 ocean, 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 far enough North to again be opposite Antarctica. There are a number of Google Earth-based toys demonstrating this, including [[http://www.zefrank.com/sandwich/tool.html "If The Earth Were A Sandwich"]].
70% of the Earth's surface is water. Mostly, that's where you end up.
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 [[http://en.wikipedia.org/wiki/File:Antipodes_LAEA_inverted.png this map]] or [[http://en.wikipedia.org/wiki/File:Antipodes_LAEA.png one like it]].)
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 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.
Now where's that shovel?