I didn't start finding things amiss with RE, it was the moon landings.
When I first heard people saying the moon landings were faked, I thought "Okay, I guess they could have been." But the more I looked into it, the less likely it seemed that they had been faked. When I saw original footage compared to all possible methods of fakery, that was the clincher. There was just no way it could have been faked. As far as I'm concerned, the moon landings were for real.
When I heard of the Flat Earth Society, I was a little more skeptical of it, because I've traveled a bit, and seen stuff that doesn't fit in with any kind of flat earth theory. Still, I thought I'd hang about and see if there was any credible, compelling evidence to support the idea of a flat earth. There have been a few ideas that looked good at first glance ('bendy light' had the best chance of explaining the difference between what I see every day, and what FET says is reality), but so far they have
all broken down under further scrutiny, where the round earth explanations have not. So, as far as I'm concerned, the earth is round. I'll keep an open mind on that one though, in case something comes up that actually does explain how the earth could be flat, and still allow for the observations I have made.