Separating eggs gives me fits. Invariably I get hasty and accidentally end up w/ yolk in my whites. That spells R U I N. Egg whites will not form peaks if even of trace of yolk or shell remains.
Here is how to do it with minimal risk. First wash your hands and put next to you a damp cloth or paper towels for your hands.
Set out three bowls: 1 to hold the whites, 1 to hold the yolks, and a smaller one into which you will separate the whites . Crack egg gently. Then open the egg into one of your hands. Gently spread apart your fingers and let the egg white spill into the small bowl. Once most of the white is in the bowl, put the yolk in the larger bowl reserved for yolks. Scrutinize the egg white for shells or drops of yolk. When satisfied that your egg white is perfect, pour it into the larger bowl reserved for egg whites. Do this to each egg. Now the beauty of this method is that you can screw up one egg white without ruining the previous 5 you've separated successfully.
If you want, use an egg separator. I find those slow. You can also separate the egg by juggling the yolk back and forth between the halves while the white drips down. I can't do this. Shells are sharp and easily break the yolk. Fingers generally do not have sharp edges, and while it is a mess, it works.