I see pictures. Pictures with horizon curves. Then I take pictures. Check to see if the curve matches what's in the first pictures. There was a resemblance, and so I proved to myself that the world is round
I go see a rocket launch. I read and see pictures and watch documentaries about space. I research the Apollo program. I see the pictures they took from the moon. I believe the earth is round.
Two ways many people here probably assure themselves it's round. While the space program path is usually more faith-based as you yourself haven't done it, if you don't believe in a conspiracy, then it would be perfectly viable. It's the same way as FEers do it.
I looked outside. I saw a flat horizon. I took pictures. I saw no noticeable curve. Resultant curve can be produced by a faulty lens and is small enough to ignore. The earth is flat.
I did some research and read some books by famous scientists (or anyone else, etc.). I myself performed some of their expirements and got somewhat similar results within a tolerable range. The earth is flat.