Sounds like it should be easy, but if you want something in particular, there are things to know.
Put the egg(s) in the pot before you put the water in, otherwise some may crack. Cover with water. If an egg floats, it is bad. Take it out and throw it out.
Put pot on the stove on High. When the water starts to boil, start timing. The length of time you actually boil the egg determines your outcome.
3 minutes = cooked, slightly loose white + liquid yolk
4 = firm white, yolk liquid in the center with firm outside edge
5 = firm white, yolk half solid with a liquid center
6 = firm white, yolk cooked through
>6 = white starts to get rubbery and the yolk begins to green on the outside.
When the pot has boiled for the desired length of time, take it off the stove, pour the water out and replace it with cold water. Add a couple of ice cubes to cool it faster.
If the eggs are very fresh, they need to be cool through to peel cleanly. If they have come from a conventional grocery store rather than your own chickens, a farmers’ market or a co-op, they will peel easily no matter what the temperature. The fresher the egg, the less easily it releases from the shell. Very fresh eggs that are intended for devilling do well to sit in the refrigerator over night.