You can always cheat and purchase a plastic egg-separator. If you don't think it's worth it, follow these simple steps to separate your egg yolks from their whites:
Crack the egg lightly, as close to the middle as possible, being careful not to break the yolk.
Hold half the shell, with the yolk in it, in one hand, and the other half-shell in the other.
Allow as much of the egg white as possible to run out of the shell and down into a bowl, then transfer the yolk to the other half of the shell, in the process allowing more of the white to run out.
It should not be necessary to perform this transfer more than twice to get the yolk almost completely clear of egg white. The process is a little awkward, however, and takes some practice. It is a good idea to have extra eggs on hand, in case the first few tries are less than successful.
If you want to keep the yolks, it doesn't matter very much if one breaks. If you want to keep the whites, it is necessary to be much more careful, and use one bowl in which to keep the whites already separated, and a different bowl over which to separate the current egg. A little white in the yolks doesn't matter, but a fleck of egg yolk can make a batch of egg whites unusable because the fat in an egg yolk will stop the whites from fluffing up when they are whipped.
Egg whites can be saved by freezing. It is not, however, worth the effort to try to save egg yolks.