Flat Earth "Theory" is a really interesting example of a pseudoscience, because obviously the proponents have jumped off the deep end. Their arguments however, are very similar to many other brands of pseudosciences. This makes it an excellent example of things you can look out for when you examine claims.
The first thing that should make you question any idea is when it requires the rewriting of all science textbooks. If the Earth is flat, you need to come up with new explanations for gravity. The usual explanation for this is that the Earth is accelerating at 9.8m/s2. The problem with this is that as we moved faster and faster, you would need more and more energy to keep accelerating the earth (this is one of the effects of relativity). Flat Earther's also believe the Earth to be 6000 years old. A constant acceleration of 9.8m/s2 for 6000 years would mean that the Earth would now be traveling a significant fraction of the speed of light.
Another red flag for any idea is the necessity of a mass conspiracy. For the Earth to be flat and not have it be common knowledge, you need to assume that every government in the world is working to keep it quiet. You also have to include all astronauts, all of NASA, Lockheed Martin, Boeing, high altitude pilots, the list goes on. When you have so many people involved in such a large conspiracy, you have to question why we don't have tons of whistle blowers. This is usually explained away by saying that the government threatens people of pays them off. That may work within a small group of people, but do you really think that with hundreds of thousands of people involved, not even a small number of them would be speaking out?
It is easy to laugh at someone who thinks that the Earth is flat. We might call them a stupid or a kook, but we must be careful. The idea that the earth is flat is ridiculous. Very smart people can however, be brought to believe very stupid things. Unless we make a concerted effort not to, it is easy to believe convincing rhetoric. This is why we must ask for evidence before accepting a claim. Skepticism is a tool that allows us to tell the difference between a flat earth and the wondrous claims of science.