1. Know what peer-review is, and isn't.
Peer-review is a good thing, peer-review is not a perfect thing. Most importantly, peer-review does NOT mean that someone repeated your experiment and verified your results. What it does generally mean is that a few people in your field read your paper, did not find any flaws in your analysis, and thought it worth publishing in whatever journal you are considering. That's a good check, but it's not perfect. Bad research still makes it through the peer-review process, and plenty of good research is published without it. There is also some movement away from peer-review today because, one, it takes a decent amount of time to go through the process, and two, there is enhanced concern to make research "open-access" so anyone can see it (for free), and many traditional peer-reviewed journals are not open-access.
2. Good thinking is in the details - look for details and data, not just conclusions.
Perhaps the best thing I can do here is offer an illustration: "Poll finds East Lansing voters not supporting proposed income tax hike". That article isn't exactly hard science, but it was of local interest to me, and it makes the point.
East Lansing (where I live) was considering enacting an income tax. Per that article, a poll found that 41% of voters favor the tax, 32% oppose it... and there is no additional information. When I read an article like that, I am automatically looking for details, like - how many people did you survey? How did you survey these people? Standing on a sidewalk downtown might get you a very different audience than randomly calling landline phone numbers. Exactly what question were they asked? We can make a pretty good guess at that one, but we know that slight variations in how a question is asked can significantly affect the response. I want those details, and without them I have no idea how much to trust the final numbers.
Always look for the details - and unfortunately, that's a skill you only get better at with practice.
3. If something ever feels wrong, or you are skeptical, try to find the original study.
In my experience, it is amazing how often media reports will greatly misstate the claims actually made by a scientific study, and that includes "science news" websites and social media accounts. If something ever feels wrong, there is no substitute for clicking links and digging around and trying to find the original journal article the media reports are derived from, and just reading the original journal article instead. You may indeed be surprised by just how readable it is... or, you may find it quite difficult to understand, which might help explain why media reports (most journalists not being scientists themselves) are so often wrong in the first place. But there is no substitute for finding the original study.
4. Finally - know what science is, and isn't.
Science is a system for describing how the world *is*, it is not a system for describing how things *ought* to be. To put this another way, naked science by itself cannot make a single moral judgment, although it can certainly help with moral judgments in a secondary sense, by presenting and organizing relevant data. Any time someone makes a statement about the way things *should be*, in a moral sense, they are at a minimum bringing some assumption in addition to the scientific data to the table, and we should be aware of that. To quote Tim Keller by example, "Most of the things we most deeply believe in - for example, human rights and human equality - are not empirically provable."