I just watched the documentary "Behind the Curve" on Netflix. It explores the Flat Earth movement and their efforts to prove that there is a massive global (or discal?) conspiracy to conceal the fact that the world is actually flat, and may or may not be covered by a dome.
This isn't about Flat Earthers specifically, but they illustrate something that's important. You'll notice that there are no Flat Earthers who manage satellite networks or who plot flight paths for international travel. The reason is obvious, the roundness of the Earth is critical information to being able to do these endeavors successfully. If you try to do these things based on the assumption that the Earth is flat, you will fail. Your satellite will crash and burn. Your plane will not get to where you want it to go.
You would think that would be the end of the story, but of course it's not. Launching satellites and plotting flight paths are highly technical skills that require years of study and training. As a result, these things are very disconnected from most people's lives. If you're, say, an accountant, you can happily believe the Earth is flat and lead a full, happy life with no negative consequences whatsoever. On the other hand, you don't see a lot of accoutants who believe that their Beanie Baby collection is tax-deductible. Ones who do would get very quickly disabused of that notion. The point here is that when you're insulated from the consequences of your beliefs, it's easy for them to stray from reality.
This leads us to a useful heuristic when evaluating claims. When someone tells you to believe that X is true, you should ask yourself, "What have they built that depends on X being true?" If the answer is "nothing", well, that is not conclusive proof that the claim is wrong, but we can relegate it to the realm of a hypothesis. It is, at best, an untested assertion. Rely on it at your own risk.
Conversely, if you would seek to convince others of your own claims, the way to do so is not to argue with them. Simply act on what you believe to be true, and show the truth of it through your results. You can argue with a flat-earther until you're blue in the face, but they'll never change their mind. In the meantime, you can build a global transportation network and satellite communication system, and not a single one of their objections will stop those from working.