I don't think "teach a man to fish" is a reasonable goal. Evaluating the literature presents many problem to laypersons.
First, most literature is behind a paywall. The abstracts are helpful, but I've read a fair number of abstracts that don't even mention the conclusion!
The easier places to get even sciency-sounding information on Googling are the blogs, and definitely the Mayo and similar sites. More often than not, the blogs are wrong. I often read the Mayo (etc.) sites to see what my patients were reading after giving them a diagnosis, even though I was "one of those doctors" who told my patients not to Google their diagnosis, especially if it was a scary one.
Another problem is how to get around the medical "jargon". I say "jargon" in quotes because though it may have started out to exclude the non-medical people, I don't think that's why we use it today. In any case, it looks like jargon, and it is intimidating to the uninitiated. We can't teach users every term they're going to meet; even the easy ones ("costochondral margins") intimidate. We can tell people to Google terms they're unfamiliar with, but it's easier to just ask someone who might know already.
Some papers are better than others. Some are so hard to decipher that even scientists misinterpret what they are saying. A great example of that was the Gur et al. paper, Sex differences in brain gray and white matter in healthy young adults: correlations with cognitive performance. It was so tech-heavy that the author him/herself actually misinterpreted their own findings data in the discussion section! Talk about badly written! And the press picked it up like free money. "See? Men are more analytical than women! There really is a difference! This is why men are better at every occupation on earth that requires reason!" And, sadly, "See? God created men to do the thinking! Women should obey their husbands, because men have more grey matter than women!"
Learning how to find and evaluate the literature is a big part of our medical education, because once we leave the umbrella of "attending physicians", we're on our own, and that means we have the literature. Conferences help, consults help, but the key is, unfortunately, the literature. There are plenty of docs who themselves cannot evaluate the literature as well as they should. I count myself among these docs, so I'd like to think there are a lot of others out there like me! When I read a paper, sometimes it's a lot of work for me. A well written, thoughtful meta-analysis is a joy to read. That Gur et al. paper, I had to read that three or four times to figure out why my gut was telling me it was wrong. When I finally understood the findings and figured out that the authors themselves misrepresented their findings, damn, that was 5 hours of my life I'll never get back.
So, the idea of teaching lay people how to evaluate the literature is problematic for me. Maybe you can do it. My intuition says not. Again, it's much easier to just ask someone who knows.
The answer would focus on the quality of medical studies (meta analysis/systematic reviews, RCTs, retrospective analyses, etc) and the reporting source (peer-reviewed original research, non-peer reviewed research, secondary sources, anecdotal self reports, etc) and how to efficiently search the medical literature and evaluate the conclusions.
I can only imagine this, and it makes me smile. I think of some poor person with one tab opened to this canonical answer, reading the paper and asking themselves, "Ok, is the Journal of Pharmaceutical and Biomedical Analysis a peer-reviewed journal?" (Googles that.) Next. Is this original research or is it a meta-analysis? :scans paper: "Looks original. Now, was the size of the study large enough?" (finds out N = 60). :thinks, hmm, I don't know if that's enough. It seems like enough.: "Well, is the control group a good one?" :um... um... I'll just keep going.: Etc.
I just don't think that's going to happen, though the thought of it actually happening does bring me joy. If only! And if we can teach laypeople to do this, can we have Philosophy.SE, Politics.SE, and Cog.Sci.SE have canonical posts about logical fallacies and cognitive biases so people can read the news and books with newly opened eyes? OK, I'm going to basque in this wonderful, joy-inducing new world for a while and get my endorphins up. Cuz, "Look, if you had one shot, or one opportunity/ To seize everything you ever wanted. one moment/ Would you capture it or just let it slip?/ Yo..."