The pan is critical for easy egg cooking. If you have to, go to the store and buy a brand new teflon pan. A heavy bottom is nice, but you don’t need a high end, ultra durable pan. If you cook only eggs and other light fare in the pan, never using metal utensils, it will last nicely. Don’t fry hamburgers in this pan.
The thing about eggs is that they are sensitive to high temperatures. If you have made scrambled eggs in a puddle of water, it’s because you cooked them too hot, the proteins squeeze together so tight it forces the moisture out of the eggs.
For scrambled eggs, whisk your eggs together in a big bowl with a bit of water, maybe a tablespoon for 2 eggs. Whisk them well and add your salt and pepper, and any other seasonings you want. Heat some butter in the pan (as much for flavor as for non-stickness) when the butter is bubbling hot, add your eggs. Now, don’t go away, you stay with the eggs until they’re done.
You should be able to see the eggs cooking while you stir, but they should not be cooking fast, let it cook at a moderate pace. As it starts to tighten up, lower the heat so you don’t overcook. When they’re close to “fully” cooked, but are still moist, get them out of the pan, they’re done. Any additional cooking will happen on the plate. You’ll get nice, soft, fluffy scrambled eggs this way.
For omelets, one other way to dish it up is to flip the omelet out of the pan right onto the plate. It goes like this, instead of flipping half the omelet over in the pan, you flip over a third of the egg, right from the handle side of the pan. Shake the pan to ensure the egg is moving freely. Then you grab the handle backwards, with your palm up and your thumb pointing away from the pan. You can then tilt the pan down so the far end is right at the plate, shake the omelet so it lays partly on the plate, and flip the rest of the omelet over the top. It’s pretty slick and neat when you get the hang of it.
Another dish I like is egg-cups, sort of a mini-quiche. You whip up a bunch of eggs with chopped veggies, meats and cheese of your choice, then put the mix into a muffin tin and cook in a low oven until done. Use a teflon tin, foil inserts or grease it really really well before cooking. I wore out an entire brillo pad trying to get egg residue out of an aluminum muffin tin.