One of the things I remember is that when you've got some young chickens and they're just starting to lay eggs, often some of the eggs don't come out quite right, so you end up with really small eggs (like half as big as normal), or big eggs with two yolks, or even eggs that have no shell! (The white and yolk of the egg are surrounded by a translucent sac, so the egg still is recognizably an egg, but it's completely soft.)
We mostly mail-ordered the chicks, but on one or two occasions we hatched eggs in an incubator. You'd turn the eggs every so often (the eggs were marked with a pencil so you could tell if they had been turned yet) and you could tell the progress of the chick by shining a bright flashlight at one side of the egg while you looked at the other (the technical term was 'candling', I think). Then eventually if all went well the chicks would hatch. Very cute!
We also had goats for a little while, but my memories of them are less fond, since one of them would always try to eat my hair.