You’ve got to get them talking. Don’t explain to them, ask them to explain to you. If someone demands evidence of climate change, you ask them to explain the evidence in favour of climate change. When they can’t or won’t, you then get to explain to them that a person who demands evidence but doesn’t seek it out is being willfully ignorant. How do you know the evidence is wrong, if you don’t know the evidence? Etc.
Stepping back from the argument at hand to criticise the person’s way of thinking is useful, but do try not to drift into ad hominem. “Ill informed” is good, “stupid” is not helpful. Be prepared for people to try to put those words into your mouth too, as in “you think I’m some kind of idiot!” Stay focused. No, I don’t think you’re stupid at all, but you’ve admitted yourself that you’re not well informed on this particular subject. It will help a lot if you do actually have some respect for the person you’re arguing with.
Speaking of respect, active listening is a great way to really engage someone beyond just shouting your arguments at each other. Ask questions, and repeat their answers in your own words to show that you’ve understood. Sometimes you don’t need a structured argument in five parts with introduction and conclusion. Sometimes you just need to get their side of the story, show that you understand it and that you are sympathetic to their point of view, and then you just say “nah. I’m pretty sure that’s wrong.” Hopefully they’ve given you enough information that you can explain in simple terms the problem with their argument, and the fact that you really, honestly heard them out and you still don’t buy it is much more disquieting than to tell them that they’re wrong from the outset. Maybe you’ve heard it all before, but maybe not, and if you’re really going to set about convincing someone that what they think is wrong then maybe you owe them the courtesy of finding out exactly what it is that they think.
If someone is truly wrong, there’s always an inconsistency to be found by asking questions. If someone believes the Bible over their own eyes, ask how they trust their eyes to read the Bible. If you can’t find the inconsistency, you may need to reassess your own point of view. Always be willing to accept that you could be the one who is wrong, however unlikely it may seem to you.
Remember that people are prideful, and our culture places a penalty on admitting fault. Don’t expect a stubborn person to say “gosh, you’re right,” but if you can show the fundamental dissonance in their thinking and suggest a better alternative, they may quietly come around. In my experience it takes about a fortnight for someone to forget the argument and decide that they’ve had the revelation all by themselves. You don’t get to say I told you so, but that’s not what you’re in it for, right?
No you didn’t.