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.
you made my day

Go to Trump and ask him what he believes. Then wait five minutes after his reply to update and counter his stance. Then wait five more minutes ... Or, ask Hillary, whereupon she'll just blatantly lie and have you knocked off if you become a threat to her existence (if you're lucky, you'll get to choose the park you desire to have your suicide body found).