The far ends of the earth

Antipodes.

In geography, the antipodes (from Greek αντι- [anti-] "opposed" and πους [pous] "foot") of any place on Earth is that point which is diametrically opposite to it. These two antipodal points are connected to each other by a straight line which passes through the very center of the earth.

In Britain, "the Antipodes" is often used to refer to Australia and New Zealand (and "Antipodeans" for their inhabitants), despite the fact that neither Australia nor New Zealand actually overlap the antipodal points of Britain. However there are as far away from the London as any land mass on the earth can be.

So, for example, one can draw a line starting from Perth, Australia, through the center of the earth and burst through the crust in the middle of Hamilton, Bermuda.

Other antipodal points include:
Auckland, New Zealand and Gibraltar
Buenos Aires, Argentina and Shanghai, China
Madrid, Spain and Wellington, New Zealand

I've been to Shanghai, so I ought to go to Buenos Aires. I've transfered planes in Auckland (which is the further point south I've ever been on the globe) and I have always had the bug up to see Gibraltar. It's just another far flung dream I have to hit these stops.

If you want to know more about antipodes, see this antipodes calculator and wikipedia's antipodes map.