the peer review side is the big unknown part, sure, but that's really about stopping the wild claims and poor quality science from making it through. doesn't always work, admittedly, but it's better than leaving it up to the press. the main issue with not understanding peer review, imo, is when you tend to assume that someone speaking on tv as a scientist is inviolable and correct, even if other scientists disagree and/or the science has not yet been peer reviewed so they're basically expressing a personal opinion. unless what they're saying is published in a peer-reviewed journal and has been repeated elsewhere, it's conjecture without evidence.
imo, we should be teaching some of the philosophy of science as well, though - that's where you'll understand that the underlying principle is falsifiability - you're looking to *disprove* rather than to prove. scientists basically are trying to get results to disagree with the initial hypothesis, because the only thing you can do with absolute certainty in science to is disprove something. that's why we talk about theories instead of laws in science - there's an implicit admission that a better theory might explain it in the future. all we've done is come up with a model that we find useful. personally, i found studying the philosophy of science (a level, and further degree-level courses on it during my physics degree) gave me a much better understanding of why science does what it does than doing a physics degree and getting papers published during my summer job in a lab at DERA.
the other things we probably ought to teach are critical evaluation of sources and basic logic. far too many people seem to have no understanding of either, and will take wild statements from someone famous without evidence as fact and won't think through the logic of vacuous claims.