I would suggest that a canonical Q/A or three could cover
(a) The basic meaning of the modulus $a=b+km$ equivalent to $a\equiv b \mod m$ for integral $k$. This basic observation is enough to deal with some questions.
(b) Solving a linear congruence to prime modulus, with examples
(c) Solving a linear congruence to composite modulus, with examples
(d) Solving simultaneous linear congruences in a number of variables to the same modulus, and relationship with matrix solutions and elimination solutions to such systems, which may be familiar in the case of $\mathbb Q$ or $\mathbb R$
(e) Solving simultaneous linear equations in one variable where each congruence is to a different modulus (the Chinese Remainder Theorem) - including examples from puzzles (eg when do all the cyclists meet again. all the clocks strike at the same time - when they are going at different rates) since some questions will arise from such puzzle contexts.
It is hard to pitch the level though, because some of the questions asked are very basic, and therefore beginners level will be necessary for some aspects. So I would suggest answers be invited at two or three different levels, highlighted in the answer as basic, intermediate or more advanced.
One issue at the more advanced level is that questions at that level sometimes focus on a particular detail which is not understood - and a generic answer might not quite serve the purpose. So there would be a case for leaving such questions open until that detail is teased out and incorporated into a canonical answer, or a comment on it.
One editable answer should consist of references and links for further reading. The relevant chapters in Hardy and Wright "Introduction to the Theory of Numbers" cover a wide range of material about congruences of different kinds. And that is just one example.