How as an individual can I know if the Earth is a sphere or a flat disc? What experiment can I do that doesn't involve trusting information from a 3rd party that would prove what the geometry really is?

Go to a church, ask them what they believe. The truth is usually the opposite.
LOL you got me, just spit out half of my dinner all over the keyboard (damn you Vod) ... but it's a good point and I agree.