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 The Synagogue of Satan, ask them what they believe. The truth is usually the opposite.
Oh! Man who has bewitched you to ignore the solemn call of thy creator. Seek him while he yet maybe found of thee.
Go to a church ask them what they believe. The truth is usually what they tell you.
I hate to say it but he's actually correct about that. Go to a Catholic church and ask the priest if he believes the Earth is a spinning ball going around the Sun; they're all yes men.