I keep a bottle of diluted bleach on the kitchen counter with which to spritz and disinfect the egg. Sometimes the eggs will have a little dirt, straw, feathers, or poop on it so we are always careful to wash them thoroughly after which we rinse it well and dry it off.
On the top of the egg we use a pencil to date it. Simply "6/7", or whatever date it is. This is easy enough for the kids to do and understand. It is important to put both the month and day, especially if you have several chickens, because your egg supply will build up and you want to use the oldest eggs first.
I keep several cardboard egg crates, just the usual supermarket type with the flip top. On one end of the crates I have written "new eggs" in big red letters. On the other end I have written "use first" in big black letters. The two crates sit side by side in the refrig. After new eggs are collected, washed and dated they are placed in the "new egg" crate filling it from the back to the front (oldest eggs in back). Then, when a crate is full you turn it around to the "use first" end of the crate. The oldest eggs are already right at the front of the crate.
Since we have two chickens and there are two sides of a typical dozen size egg crate we usually keep all Lucy's eggs on the left and all of Rosie's eggs on the right. This way we keep track of who is laying and when. We can tell the two apart because Rosie's eggs are smaller and darker in color than Lucy's.
Lately one of the hens, Lucy, stopped laying eggs and wants to sit on the nest all the time. She has become "broody". I have done lots of research online trying to find some way to snap her out of it. Some of the suggestions were locking her out of her nesting area, putting her in a cage for a day, dipping her butt in cold water. I only tried locking her out of the nesting area--the other's seemed mean--it didn't work. She built a nest in the yard and sat there all day. I guess I'll just leave her alone and hope she comes out of it soon; my egg supply is running short.