Ah, the egg!  So simple and so easy to mess up!  Under or overcooking these suckers can be disastrous!  Eggs are used in so many different ways, but a lot of the principles are the same.

Let's start with using them in baking.  The eggs you use should always be at room temperature!  Your baked item will not be as fluffy if the eggs are cold.
Poaching.  Poaching takes time and practice!  It's not that hard once you get the hang of it!

You want to get a pot of water gently boiling.  That means it's barely boiling, not roaring.  Put about a tablespoon of white vinegar in the water.  Crack the egg in a bowl and mix the water in the pot in a circular motion to create a soft current.  Gently slide the egg into the water.  Let cook about 3-4 minutes and carefully remove with a slotted spoon.  Now fresh eggs make the best poached eggs, so check the date on your eggs and make sure they are on the fresher side.  Since there is not oil used, poached eggs are very healthy.  They are mostly eaten on toast or in eggs benedict (on an English muffin with ham and hollandaise sauce)
 

Scrambled.  You may already be making scrambled eggs on a regular basis, but this is how fancy restaurants are doing it to get them nice and fluffy!  First crack your eggs in a bowl and whisk them together with some salt, pepper and about a tablespoon of milk (or cream if you want to make them French style).  Put a pad of butter in a small pan on medium low heat (never cook eggs on high heat!!  this will make them dense and rubbery!); when it has melted, swirl it around to cover the whole pan.  Slide your eggs in and agitate.  Agitate meaning whisk them constantly.  Continue to whisk and take the pan on and off the heat to control the temp.  Once they have finished they will look a little more like oatmeal than scrambled eggs, but they will taste like rich, fluffy heaven.  If you have never tried this method, go for it!

Hard Boiled Eggs.  These are pretty easy to do, but again the key is to not over cook them.  If you over cook them you can get a green ring around the yolk.  This is not mold, but it's actually a build-up of iron and sulfur from the egg over cooking.  If you have a hard boiler machine you are in luck  and these are easy for you, but doing them in a pot is a little more difficult.  An egg boiler machine looks like this:
 

Put the eggs in a pot large enough to have one layer.  Add cold water and fill to about an inch over the eggs.  On high heat, let the water come just to a boil.  Cover and remove from heat.  Let sit for about 15 minutes. (a little longer if the eggs are really large)  Remove from the pot and place in an ice bath to stop the cooking process.  The eggs should be nice and rich inside and not rubbery.  Always refrigerate hard boiled eggs with the shells on!
Fried eggs or over easy eggs.  Start with a pan on medium high heat.  Add some cooking spray or a pad of butter.  Crack an egg and slide in in the pan.  Immediately reduce heat to low and let cook.  Cook for about 5 minutes or until the whites are solid and the yellow is firm, but not hard.  You can serve like this of gently flip to cook the other side.

Cooking eggs can be a little frustrating and it may take some experimenting, but you'll get the hang of it!