Aside from cooking eggs whole, both in their shells (hard cooked, soft boiled, and coddled) and out (poached, fried, and au plat), most preparations that utilize eggs will require either beating yolks and whites together, or separating the yolk and white and beating them separately. Here are some tips for the best methods to use when performing these basic but essential tasks.
Many recipes call for eggs that have been separated, which means they are divided into yolks and whites. It's a good idea to give the eggs a sniff test before you start a bad egg will be obviously smelly. One bad egg will spoil all the rest.

Have two nonreactive bowls ready. Tap each egg in turn against a hard surface (the lip of the bowl is good for this) to break its shell. Insert your fingers in the break and halve the shell roughly along its "equator," keeping the egg's contents in one shell half.

While shifting the contents back and forth between the shell halves, allow as much white as possible to fall into one bowl. Finish by dropping the yolk into the second bowl and discarding the shells.
For egg whites to be beaten successfully, it's important that they be free from even the smallest speck of yolk. This is because yolks are fatty, and fat inhibits air incorporation.
If, when separating the eggs, some yolk has fallen into your whites, remove it either by scooping it out using a bit of broken shell, or by touching the yolk with the corner of a damp paper towel (the yolk should adhere to the towel).
In some cases, too much yolk will have landed in the whites. If that happens, you'll need to discard the whites and start again.

Break the eggs into a nonreactive bowl. Beat with a fork, whisk, or mechanical or electric beater until the yolks and whites are completely combined, about 45 seconds. Once beaten, the eggs are ready to be used to make an omelet or in any other recipe that calls for beaten eggs in the ingredients.

Separate the eggs into two bowls. Beat the yolks with a whisk and a figure-eight motion. This breaks them up and gives them volume. Once lightly aerated, they're ready to receive other ingredients, such as oil (to make mayonnaise, for example) or sugar (for cakes and other desserts). In many cases, the ingredients to be beaten in with the yolks can be added right from the beginning, as is the case with zabaglione.