Halving
The safest, most comfortable way I have found to slice an avocado in half is to put it on its side on a cutting board. Hold the avocado in place with one hand on top and start slicing into the fruit with a chef's knife, with the blade positioned parallel to the board. You'll be cutting along the axis that runs from the fruit's top to bottom.
Now begin rotating the avocado so the knife starts cutting all the way around the pit. You really don't need to be moving the knife much at all. Your hand moving the avocado is doing all the work for you and, more importantly, not moving ever-closer to the edge of the knife. In short order, you will have made a complete circle around the avocado. Put down the knife and simply twist apart the halves.
Pitting
Now to the pit.
I would like to once and for all quash the myth that the best way to remove the pit of an avocado is with a knife. It's nice that the New York Times recommended against digging it out with the tip of the knife, but I don't understand why it says the "right" way is by "gently striking the pit with a knife to embed its long edge into the stone." No. No. No! Sure, this looks cool, but in my experience, it's totally unnecessary. Why risk a glancing blow where the knife slips and goes into your hand? Why make this more complicated than it needs to be? Grab a spoon. Scoop out the pit. Done.
Slicing and peeling
At this point, you can do something fun with your avocado halves - bake eggs in them, for example - or simply use the spoon to scoop out the flesh. You can use a knife to score the flesh, too, for easy removal, but there's no need to run the risk that the blade will slice through the avocado skin and then into your skin. Simply use the back of the knife blade rather than the sharpened side. Or be even safer and whip out a good old butter knife.
If you don't care about keeping the avocado half intact or are looking to achieve some nice wedges, we like the method espoused by chef, cookbook author and TV host Sara Moulton. Proceed with the same horizontal cut as described above. Then, with the knife perpendicular to the board, cut the top side of the avocado from stem to bottom. Flip the avocado over, and repeat on the other side. You end up with four sections; the pit is attached to one, and you can easily pluck it out with your fingers.
The sections also make it easy to peel the skin and keep them whole. Just run a spoon between flesh and skin. If your avocado is really ripe, you might even be able to get away with peeling the skin off with your fingers.
Saving
If you want to save an avocado you've cut without it oxidizing and turning brown, you can apply something acidic to the surface, such as lemon or lime juice or vinegar, before wrapping it in plastic. The browning doesn't matter much to me, so I usually just tightly pull plastic wrap over the flesh, and sometimes that is enough to keep discolouration at bay.
Whole unripened avocados can sit at room temperature until they ripen, usually four to five days. If you want to cut that time roughly in half, place the avocados in a brown paper bag with an apple or banana. Ripe avocados will yield to a bit of gentle pressure - they should be soft but not mushy - and may darken in colour on the outside. At that point, store them in the refrigerator for up to two or three days.
Now that your crash course is complete, the only danger you'll face from avocados is eating too much guacamole.