How can I remove the peel and pit of an avocado without the whole thing turning into mush?
The California Avocado Commission recommends this (safe but wimpy - see below for a better way) three-step process:


Start with a ripe avocado and cut it lengthwise around the seed. Rotate the halves to separate.
Remove the seed by sliding the tip of a spoon gently underneath and lifting out. The other common seed-extraction method - striking the seed with a knife and twisting - requires some skill and is not recommended.
Peel the fruit by placing the cut side down and removing the skin with a knife or your fingers, starting at the small end. Or simply scoop out the avocado meat with a spoon. Be sure to sprinkle all cut surfaces with lemon or lime juice or white vinegar to prevent discoloration.




Source: California Avocado Commission

Notes:


Another (and my favorite) way to remove the pit is by holding the half with the pit in one hand and striking the sharp edge of the knife used in step one into the pit, then twisting to remove. This is the "pro" way to do it, but takes a little practice. The video Nate referenced in the comments shows how to do it, it's not that hard.
If you're going to be chopping the avocado up, you slice it up using the tip of the knife still in the shell, then scoop the sliced avocado meat out with a spoon.