Personally, I like multiple Foucault pendulums as this demonstrates rotation and curvature. I think the part that is missing in everyone's explanation is how to prove it is a sphere and not a hemisphere. Obviously, going to the "edge" would immediately shows this is false. But I think the only way to do this without moving too much is to record lunar eclipses and do the math to figure out that you would eventually see straight edges in an eclipse if the earth was not completely spherical.
Interesting idea! The sun would have to be about 6'000 km away I think. The full moon / new moon / solar eclipse proves that the moon is much closer than the sun. The moon would have to be dangerously close. Then, we know that the moon causes tides, and we can measure them as well on different cities. Being so close, the sun would have to cause tides as well, unless it is very light. It would have to take some time to calculate all that...
Sure, but a flat earther will just argue that the sun is much closer than scientists claim, which explains the difference in shadow length. And again, what's a layman to do?
The best experiment to prove the earth is round that I've thought up involves the Coriolis Effect and a rifle fired at different points both north and south of the equator. Unfortunately that requires international travel with firearms, making it legally difficult to try.