Further proof of the Earth being round came after the voyage of Columbus. When Newton discovered and measured the force of gravity, that number could then be tested anywhere the theory was known. Since the force of gravity is roughly the same everywhere on the globe, it could be surmised that the Earth must be spherical. If the Earth were not round, whole hemispheres would have different atmospheric pressure and significantly different sea levels. Also, pictures taken of the Earth in the last 50 years have proved absolutely conclusively that the Earth is round. These are just the arguments that don't require much physics knowledge to explain, there are others that are more technical, but I think that the simplest arguments are the best. Now that we have access to space, the easiest way to prove the Earth is spherical is to leave it and view it from a distance. Astronauts and space probes have done just that. Every picture of Earth ever taken shows only a circular shape, and the only geometric solid which looks like a circle from any direction is a sphere.
One of the oldest proofs of the Earth's shape, however, can be seen from the ground and occurs during every lunar eclipse. The geometry of a lunar eclipse has been known since ancient Greece. When a full Moon occurs in the plane of Earth's orbit, the Moon slowly moves through Earth's shadow. Every time that shadow is seen, its edge is round. Once again, the only solid that always projects a round shadow is a sphere.