To make proper hard-boiled eggs, you must start with cold water. If you're making a whole bunch of eggs at once, and merely dump the new ones in already boiling water, the eggs will tend to split and ooze. Not a good thing. You place the eggs— preferably several days old, so they will peel easily— in a pot, cover them with cold water, salt the water*, and heat until boiling. Then you cover the pot, turn off the heat, and let it sit for 10 minutes. (PLEASE NOTE: That time changes with altitude; consult a high-altitude cookbook for proper timing.) Run cold water over the eggs in the sink until they are cool and safe to handle. There! Hard-boiled eggs.
*Salt in the water helps the egg to release from its shell.
But then you have to dye them. Dye kits are inexpensive and widely available, but you can make your own if you really want. It takes about 2/3 cup water and 2 tsp. distilled (white) vinegar, plus as much food coloring as needed to get the right shade. Celestial Seasonings and Home & Garden Television have recipes for natural dyes. Celestial Seasoning also mentions that adding a little vegetable oil to your dye will create a marbelized look.
If you're really hard-core about decorating eggs, you can do Ukrainian style eggs, also known as pysansky. Be aware that not only is this a time-consuming process (with the traditional tools including a candle; no doubt professional decoraters have nifty technical tools), but that a properly made pysansky is done with a raw egg, so it's dangerous to handle for several months (until it dries out entirely.) Simpler effects can be achieved with the wax crayon included in most decorating kits; leave the crayon in the sun for easiest use.
Okay. You've boiled them, now you must eat them. Last week's egg salad is one way to get rid of the suckers. Deviled eggs aren't that different of a concept, but take a little more work. They are so popular that there are platters designed just for serving them.
For deviled eggs, you always start with the basic hard-boiled egg. Most recipes I've seen ask for six, so assume all proportions work from that number. You peel the shell off, cut it lengthwise, and remove the yolk. The yolks go into a bowl with varying ingredients, depending on your recipe.
Goodegg.com has a classic recipe where you mash the yolk with 3 tbsp mayonnaise, 1 tbsp sugar, 1 tsp mustard, 1 tsp vinegar, salt, pepper, and paprika. Okay for the basic idea, but you can replace the mayo with mustard (or dijon) for a kick, or go in a new direction entirely, adding horseradish and Parmesan cheese for a new take, jalepeños, chiles, and cumin for a Southwestern feel, or you could get very swank and use a recipe that requires making the mayonaisse from scratch. Neat.
If none of those recipes does it for you, do an online search and you'll turn up thousands more. Or no doubt, the other participants of the Carnival of the Recipes will be happy to share their personal recipes.
Once you've mashed up all of the ingredients, you have to put them back in the egg halves. If you have a deft hand with a spoon, this isn't a problem, but there's a more fun alternative. You take a large ziploc bag, put the mash in there, and seal the top, then cut a small hole in one corner and squeeze the stuff out like it's a pastry bag. (You can, in fact, use a pastry bag.)
So then comes the question, Why am I pointing to all of these online recipes instead of sharing my own? Well, when I was a kid, I hated hard-boiled eggs for a while. I think it had to do with choking down a whole egg when I was sick, not precisely the stuff of good memories. I didn't develop an appreciation for deviled eggs until I was in college, by which point I didn't have a family recipe to fall back on. I suspect my recipe would feature mustard and some onion variant (like chives), but I'm not willing to inflict an untested recipe on you.