The ancient Greeks did it, and you can too. Find out how some easily observable facts allow you to measure the approximate distance from the Earth to the Moon.
One of the hardest parts of calculating distances in space is the difficulty finding reference points. The size or distance of objects on Earth can be hard to estimate, but they occupy a landscape which can be measured, providing a jumping-off point. The moon gives up a few clues — it's clearly closer than the Sun or the stars, but it's still drifting in a nothingness that's hard to measure.
The distance to the moon was measured, or at least approximated, over 2000 years ago, by our old friends, the Greeks. They'd already figured out the circumference and consequently the diameter of the earth, providing the one absolute number on which to base the rest. After that, it's geometry.