Serious rationality training would involve:

• Studying the relevant concepts that describe the process of reasoning, such as "argument", "reason", "definition" or "concept". This allows to talk about thinking, and to study what counts for rational or erroneous thinking (reasoning, arguments).
• Studying the tools for making thinking more rational, recognizing or avoiding errors in thinking, evaluating correctness of thinking. For example, taboo a word (make sure you have an understanding behind it), focus on a question (while completely stopping from making arguments about the previously discussed question in the context of which the new question appeared), explain which specific reasons (assumptions) about the discussed concepts your argument used.
• Having someone to point out errors in your own thinking. This is like learning language, or good manners: you make errors, they are pointed out, you learn not to make them in the future. As a bonus, the rules are not arbitrary, and serve actual goals, with understanding of this connection being part of the art. For example, you argue for a correct conclusion using a correct argument which in itself is not sufficient to come to that conclusion - this could be a very subtle error that needs pointing out.

The first two items of this process are reasonably accomplished by the Sequences, but of course the growth of cultural knowledge would be better facilitated by people creating actual curricula or ideally writing up their own lecture notes.

The last item, practice, requires a trained rationalist and a topic to discuss, any topic where we have sufficient wealth of concepts to trip over and insufficient understanding to have answers that are not already cached to correct values and need actual reconstruction.