I think everyone should start with an understanding of the scientific method to get a basic understanding of what science actually is. I'm pretty sure that at least can be understood by almost everyone.
I think once people understand how science works, then they can start to assess scientific claims in a more effective way.
The problem is that many people think they understand how science works,
even when they don't have scientific training or the background knowledge to correct their own misconceptions.
It is frustrating to whom people with whom I agree politically about the importance of global climate change make absurdly wrong scientific claims in support of their position. The opponents of global climate change pick holes in these misconceptions... and they are right to do so. However, it takes away from the ability of real scientists to make real scientific claims that are actually valid.
There is also the problem of simple intuitions, slogans, that simplify scientific concepts and are correct (scientifically) but widely misunderstood.
Everyone knows (and almost everyone believes) that nothing can go faster than the speed of light. But very few people know what this means. I can easily ask a couple of questions to non-scientists to get them to make statements about the speed of light that are wildly untrue. People think they understand it. But, they don't.
Science is by nature unintuitive (in the sense that what science says is very different than what people would expect). If science were intuitive... you wouldn't need science (things would just work the way people expect.
When you study science, you learn to question your intuition and to look for facts that contradict your assumptions. It happens all of the time in science, and when beliefs are challenged by facts... in science the facts always win.
Many people miss the point that science challenges assumptions. Instead they use science as just a set of new assumptions. Without scientific training... things like "nothing goes faster than the speed of light" just becomes a bumper sticker, a new assumption. Worse, you get axioms like "nature abhors a vacuum" and "there are no straight lines in nature" that sound scientific, but or not.
In order to be scientifically literate, you need some way to distinguish between a "basic understanding of science", and a "misconception of a scientific principle". Without scientific training, there is no way to tell the difference.