An idiom is a phrase, etc, whose meaning is not clear from the definitions of the individual words- you'd struggle to understand why someone is saying that they will eat their hat from the individual words.
A saying is a standard phrase or sentence people often use when a particular situation arises. If you know the words, you'll get the meaning of some sayings, but not all:
1 Dime a dozen
2 Dollars for doughnuts
As long as you know what a dime is (a ten-cent coin), the first suggests that something is cheap, but the meaning of the second is much less clear. The second is idiomatic, the first less so.
'Informal' is different; it means that we use the word, idiom, etc, when we are speaking in a relaxed manner. Not all idioms are informal and many words, etc, that are not idioms are informal- it just means the language we use among friends, etc.