There is so much one can do with the egg that it will be impossible to list them all, for the quest to find them all will never end, however here I will try to go over what I can.
To separate yolks from whites crack the egg in half over a bowl. Move the yolk back and forth allowing the whites to drop into the bowl, when the whites have been separated put the yolks into another bowl.
To hard boil an egg, put it in the boiling pot for 11-12 minutes, this time only counts when the water is boiling. It's best to add the eggs to boiling water and not cold water and bring it up to a boil, as it will affect cooking times and you might over/under cook the eggs. When the time is up transfer the eggs to cold water, letting the eggs cool in the cold water. They may still cook a little while in the water. When they have cooled cut one egg in half to make sure it has cooked properly, if not place the rest of the eggs in boiling water for the time needed. Next lightly smash the eggs on a cutting board and roll them with a little force, forcing the shell to crack. proceed to remove the shell, I find it's easy to pull half the shell off and then the other half, usually one half should just slide off, letting you easily peel the rest, or slide it off as well. Next wash the egg again, removing any piece of shell you may have left on.
When poaching eggs it's best to add a little vinegar to help hold the eggs together. However adding too much vinegar will cause the eggs to become rubbery.