To have grown up with an idea that earth is not flat, it never occurred to me that I will ever need to prove that to anyone, at least not to someone in my own family. But given that my grand dad (a very religious guy), who never had any formal education and who never came out of a hilly village in a poor part of an already poor country, we finally had yet another science versus religious ‘commonsense’ confrontation. He, like many other elders of the village where firm that Earth is flat, and understandably so as they never saw a world outside their village, never got any education and whatever they read was those ‘holly’ (entertaining) religious books. I took the role of a knowledgeable science educator and started my difficult expedition of proving the most important idea that governs most aspect of our life, not just of people directly concern with physics but every living creature around. As (round earth) is a guaranteed idea that we all grow up with so not many ever try to find out (or are even taught in schools) for themselves about the validity of what we believe. But during my process of educating my grand dad I realized that it is not very easy to prove this obvious science, at least not to a person with deeply rooted believes. As exclaimed in famous theorem in the philosophy of science, the Duhem-Quine theorem, states that it is impossible to conclusively disprove any scientific hypothesis. That includes such hypotheses as the earth is flat. Though it is a philosopher’s theorem whose contribution to modern day (post 1900) science is negligible, but the theorem did hold some validity at least in my case, but I was able to triumph in the end.
Proof
I had to get my granddad to travel around 200 miles with me for this experiment. I used all the basic ideas that had been used by the Greeks and many other scholars to find that the earth was round. Here they are-
1. I used the idea that departing boats gradually sink below the horizon. He saw it and faintly believed that may be it is not flat everywhere and hence it may be like a Frisbee. Admittedly this only proved that the earth is round right where you are. I wish I had a camera then to record his expression, he had seen a disappearing boat in the horizon before but never questioned the phenomenon and hence remained firm to the believe of flat earth.
Just like that apple hitting Newton, my forefathers were certainly hit by apples before and they even were curious. But they were curious for all the greedy reasons wondering when the next one will fall.
2. Second I used the stick-shadow experiment in which I stuck a stick in ground, which produced a shadow. The shadow moved as time passed (ancient Shadow Clocks use the same principle). If the world had been flat, then two sticks in different locations would produce the same shadow.
3. Third I used the constellation shift. As we know the constellations shift relative to the horizon as we move north and south around the globe (in my case just 200 miles), something that could only happen if we were standing on a sphere. For this I drew some diagrams (it’s a pity that I didn’t had an iphone during my experiment to take pictures) to convince him and this pretty much worked. This strengthened my claim and he dropped the Frisbee idea.
4. This last one just did it all for me the famous lunar eclipse (which even Aristotle used to prove the sphericity of earth. As we have seen number of times in eclipses the dividing line is always rounded. Consequently, if the eclipse is due to the interposition of the earth, the rounded line results from its spherical shape” Of course a Frisbee, properly angled, would make a round shadow too. But if the Frisbee rotated while the eclipse was in progress, the curvature of its shadow would change. The earth’s does not.
These were just the basic ideas that I could employ in my endeavor. I know there are many other ways of disproving the flat earth theories but they were not very helpful for my situation. Please let me know of any other easy ideas in the comments.
lazychemist 
This may not be feasible where you are: Alfred Russell Wallace demonstrated the curvature with 3 bridges over a straight canal ~1 mile long. Beneath each, he hung a pole with two bright markers attached, ~1 foot above the other. The distance from bottom marker to the water surface was carefully measured, and set to be the same in all cases. Then, by looking along the canal, it was clear the two top markers of the end bridges were in line with the bottom marker of the centre (i.e. that the still and ‘flat’ canal actually bulged upwards).
However, the flat-earthers he was trying to convince announced that this proved that the world was, in fact, flat. Because.
There is the Alfred Russel Wallace experiment utilizing a long canal with still water. He drove in three stake, the same height above the water. One stake at the far end, one at the near end and one in between. sighting from the top of the near stake to the top of the far stake revealed the middle stake sticking up above the level line. Wallace did this in response to a challenge from a religious fundamentalist and got into fights and was sued as a result.
The difference between geodetic earth curvature and a level line is right at 0.6 ft/mile. Flat earth theory has application over small distances. My house is built on a flat earth, for example.