The best advice I can give you is to simply avoid debates like this. You are almost certainly never going to convince a flat-earther that they are incorrect.
Beyond that, you don't really need to know that much about science to accept that the Earth is round. You merely need to trust that thousands upon thousands of scientists (and some non-scientists) are not lying when they say that the Earth is round. And that all those pictures of a round Earth are not fakes. And the curved horizon, which can be seen from an airliner, is evidence that the Earth is round. And our space probes that have been sent to other planets and bodies in the solar system are also not fakes. And a thousand other small things that make no sense unless the Earth is round.
And remember that each individual piece of evidence is mostly worthless if you try to evaluate it alone. It's only when you combine it with thousands of other observations along with the known laws of nature and try to create a working, coherent model of the Earth that all of these small observations make sense. Sure, you could try to say that ships disappearing over the horizon is just an optical illusion. Or that the Sun and Moon behave differently than we think. But you can't create a fully functional model of the Earth and the heavens, one that explains all observed phenomenon using the fewest and simplest laws of nature, without making the Earth round.
Really this comes down to whether you trust that the scientific community isn't so horribly corrupt and evil that it boggles the mind as to how it could ever exist. How could such a community function with so many selfish liars and morally bankrupt individuals? How could so many people motivated purely by self-interest ever come to a common agreement on what to present to the public? This isn't a few dozen people running a Ponzi scheme or on a board trustees, we're talking about tens of thousands of people from different countries, religions, and classes. People that graduated from the most expensive schools working with people who graduated from community colleges and schools of much lesser pedigree. What could possibly motivate so many people of so different backgrounds to do such a thing? Greed? Criminals aren't generally known to be trustworthy. Such an institution would be so filled with backstabbing crooks, each one scrambling to get that golden funding, that the entire thing would fall apart.
To believe that the explanations by the scientific community are so horribly wrong requires that one believe that they are either united and actively working to feed misinformation to the public, or that they are absolutely and utterly incompetent. The former requires explaining the above, while the latter has so much overwhelming evidence against it in the form of modern technology and medicine that it's just silly to believe.
As for how to prove that the Earth is round, just type in "evidence that the Earth is round" into google and you'll get nearly 5 million hits. The first page of results should hold a wealth of good information for you. I only send you to google because I have no better source bookmarked or saved at the moment.