Add enough tap water to come at least one inch above eggs. A tablespoon of vinegar may be added to allow better dye coverage after cooling. Let eggs stand, covered, in the hot water for 15 minutes. Immediately run cold water over the eggs, or put cold water in the pot and add a few ice cubes to hasten cooling. Dyeing the eggs works with more ease if it is done right after the boiling, vs. refrigerating them.
Coffee mugs are the ideal size and depth for dyeing eggs.
If using the drops of food coloring, get the party rolling with the basic green, yellow, red, and blue, and mix fancy colors later. Maybe someone else has a gem of an idea about this. Interesting kits seen at the store this year include stencils that are molded in egg shapes for ease of use. Eggshells are porous, and even more so after the protective coating from the processing plant has been boiled away. Storage requirements for eggs match what you'd expect of milk, meat, fish, or poultry.