This is a question I remember reflecting upon as a kid. If I dug straight down from my back yard in New Zealand, passing through the Earth's core, where would my hole come out? A new application utilising Google Earth now provides the answer.
Try it out here. If I were to dig in Clevedon (South Auckland), I would surface (ignoring the molten core issue) in Andalucia, Spain—as near as I can tell:

So, dig at home here:

And surface here (Alcazaba, Alhambra), perhaps?:

Right, who's taking first shift with the spade?









and using a globe to determine where I might come out :)
I'm in London now, so I'll have to check the reverse, could save me a packet on flying back home
oops, lost my first line...
was supposed to say: "I also remember thinking this as a kid and using a globe to determine where I might come out"