One of the best ways to learn is by doing. The introductory messages for the review queues are nice (especially now that I have dismissed them all for every site I participate on), but text isn't the ideal way to teach someone how to do a task properly. This is not a proposal targeted at brand new users to bring them up to speed. I think that there is a large group of engaged users that are experienced enough that they aren't making new user type mistakes, but that are learning how to swim by getting dropped in the deep end of the pool as they unlock more privileges and encounter less common situations on the site.
I think it would be helpful to have a library of short interactive training sessions or tutorials where users can perform tasks designed to highlight certain SE features and illustrate underlying principles and then get feedback on how well they performed those tasks. The feedback should include explicitly what was done correctly and what could have been done better. I don't think it should be a test that is scored. It should be an optional activity that lets people try out features and trains them in best practices. I think there should be some sort of badge for completing tutorials that takes into account that the number of tutorials will grow over time. This is different from review audits, because I think the content should be designed to have a clearly correct answer that highlights some aspect or feature of SE that is important to know, and not a post that has a certain disposition based on how it was previously reviewed.
For example, a tutorial for the suggested edit queue might simulate reviewing an edit that requires looking at the markdown to detect that a link has been changed to a spam link. If the user successfully detects it, they get some "good job!" type feedback and more detail about the way suggested edits can be abused. If the user misses it, they get feedback about the markdown tab, and still get the details they would have gotten if they had detected it.
Other examples: the tutorial for the close vote queue might have task to review a question where the suggested duplicate is obviously not a correct choice when you look at the answers or the flagging tutorial could illustrate situations where "low quality" should be chosen over "not an answer". The feedback could include where in the interface to look for the information to make the right decision, content guidelines, or even a little SE history or trivia to help connect users with the community (like the http:://(site)/reputation hidden feature).
I understand creating interactive training represents a lot of work, but I think it would have a lot more impact than trying to get people to read and understand text-based documentation. An investment in the infrastructure for interactive Stack Exchange training could pay off not just for training people to use their public site privileges more effectively, but also for on-boarding newly elected moderators or helping Teams users get started. This infrastructure could also provide a way to gather data in a repeatable way on how users view certain features or judge certain content, which could be used to measure the effectiveness of various education efforts.
If you wanted to go really crazy, you might make it so that SE users could design and share tutorials about using Stack Exchange sites (and chat rooms), similar to the way we can create and share SEDE queries.
I did look for similar suggestions and found Please provide some quick how to and training videos for the sites and The need to introduce beginners to the Stack Exchange Platform. What I'm suggesting is different. I don't think we need more documentation for newbs, whether it is in video, text, or interpretive dance form. I think we need interactive training for community members as a way not only to teach them how to use features, newly earned privileges, etc., but to get the community as a whole to correlate better with itself when making judgements about certain types of content.