There's actually some interesting psychological research that suggests people primarily evaluate based on how you present things: kids are only cautious when their parents seem worried, and will be much calmer and more accepting if the parents act like something is no big deal. If you present a request casually, it's more likely to be casually accepted without thought. If you seem extremely anxious, people will pick up on that and get anxious themselves. Definitely a skill I have benefited from learning.
A sub-skill I would suggest is being okay with "no". I've found that if I ask for a big favor, get a "no", and just smile and move on, then people feel safer about me in the future - I didn't make them feel bad, so they don't have to be defensive about my future requests. It also makes it much easier for me to ask for the favor, and to come off casually, because I don't have any particular investment in a "yes" answer.
Recent brainstorming sessions at SIAI (with participants including Anna, Carl, Jasen, Divia, Will, Amy Willey, and Andrew Critch) have started to produce lists of rationality skills that we could potentially try to teach (at Rationality Boot Camp, at Less Wrong meetups, or similar venues). We've also been trying to break those skills down to the 5-second level (step 2) and come up with ideas for exercises that might teach them (step 3) although we haven't actually composed those exercises yet (step 4, where the actual work takes place).
The bulk of this post will mainly go into the comments, which I'll try to keep to the following format: A top-level comment is a major or minor skill to teach; upvote this comment if you think this skill should get priority in teaching. Sub-level comments describe 5-second subskills that go into this skill, and then third-level comments are ideas for exercises which could potentially train that 5-second skill. If anyone actually went to the work of composing a specific exercise people could run through, that would go to the fourth-level of commenting, I guess. For some major practicable arts with a known standard learning format like "Improv" or "Acting", I'll put the exercise at the top and guesses at which skills it might teach below. (And any plain old replies can go at any level.)
I probably won't be able to get to all of what we brainstormed today, so here's a PNG of the Freemind map that I generated during our session.