Like a lot of things, the phrase "do your own research" started off life with good intentions, and then got co-opted by a certain kind of person to enable behaviour antithetical to the original coiner's intent. Nowadays, if someone says they "did their own research," you are far less likely to cock your head and listen intently than you are to wince involuntarily, and then seek to exit the conversation at speed.
Here's the thing – you are stunningly ill-equipped to do your own research. Unless you happen to already be a professional in the field you are researching, you do not have the education, you do not have the resources, and most importantly, you are not aware of the various tripmines and tar pits that exist in that field to trap the unwary, i.e. you. Yes, occasionally an outsider will come in with a fresh pair of eyes and see that which eluded the ossified orbs of the pros, but what happens in the overwhelmingly vast majority of times is the opposite - instead they will end up showing off their whole ass because their ignorance of some foundational shit will cause them to make a rookie mistake that those with experience would not. Then those with experience will swoop in and eviscerate the newbie and feed them to their young. The law of the jungle is harsh but fair.
When someone presents an interesting topic or viewpoint, your next step shouldn't be to jump to doing your own research on that topic, it should instead be to check that person's work. This is a subtle, but important distinction. If someone tells you that they've found out that a steady diet of honeycomb wax wards away cancer, you shouldn't immediately go and search
honeycomb wax cure cancer, because they got their theory from somewhere and there's almost definitely a couple hundred blogs with titles like "medical-info-trust.com" with plausible-sounding AI-generated posts about the topic already. You could easily spend the next four full hours finding article after article linking study after study detailing the benefits of honeycomb wax – not only for cancer, but also blood pressure, hair loss, and as a natural pesticide! Is any of it real? Who knows? Not you, certainly. You likely can't read a study to save your life, or at least not properly evaluate it, not like a trained professional with years of experience. And if you go into this research blithely, you'll likely find convincing-sounding information everywhere you look.
So should you instead become a curmudgeon, dismissing all interesting new factoids as flights of fancy and humbuggery? Well, it's certainly an option, and frankly one that grows more appealing every year, if I'm being honest. But more useful is instead to instead look for dissenting opinions. Instead, search for
honeycomb wax cancer hoax or use Snopes to see if anyone's already disproven it. And then look for arguments against the dissent! Be honest with yourself – if you're not capable of reasonably evaluating the arguments for something, can you expect to be able to reasonably evaluate the arguments against? Try to find people arguing specifically against the dissenting opinions you've read, not just new interpretations of the original media. Look up unfamiliar terms or practices they mention. Try to become familiar with the basis of why they're dissenting, which is usually a lot more accessible than the facts they're dissenting with. (A highly-useful example is knowledge of correlation vs causation, which is bound to trip you up everywhere if you don't know about it, but can improve your ability to evaluate many things if you do.)
Try to spend an amount of time on this process commensurate with how important this topic is to you, or is likely to become. Are you evaluating a major dietary change or about to adopt a whole new identity based on a radical new "truth" you just learned? Maybe first make good and sure that you know and understand the arguments against your potential new truth, and that those arguments don't dissuade you.
Ultimately, you're not just learning more about the topic in question, you're learning how to do the research. You're learning about the tripmines and tarpits that others have stumbled into. You are learning how to be objective and how to recognize traps that could compromise that objectivity. With time and practice, you'll find it easier and easier to see through bad-faith arguments with ill-supported facts, and the arguments you do agree with will be more well-supported and far less likely to cause you or others grievous personal harm. Scammers and hucksters will find no purchase in you, and fake online "medical journals" will receive no more of your advertising dollars. (That said, a good adblocker will help there too.)
Doing your own research may feel like an exciting game at times, but checking people's work is where the real value is. Though life may be a little less exciting with fewer wild and crazy discoveries finding purchase in your mind, it will also be less disappointing and/or terrifying if and when you are eventually proven wrong.