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.
To start with... there's a horizon, meaning that the surface that one is observing from is not an infinite plane. On the clearest of days, the only restriction to one's range of sight is the horizon. There can be two explanations for this - one, that the Earth at some point just stops, as if you were looking off the edge of a table. The other is that the Earth is round. Hundreds of years ago, before the invention of the compass or sextant, precise navigation for ships was difficult, even with the stars. Ships that ventured past sight of land were often lost, and thus it was generally believed that the world simply *stopped* at the end of the horizon. With the invention of the compass, and improved map-making, people began to dare more, and with the return of Columbus from his trans-Atlantic voyage, the concept of the Earth as flat was shattered.
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.
You can launch a rocket to a high altitude and take pictures of the earth (which various government agencies and private groups have already done thousands of times in the past), but that isn't the most practical way. Pictures and videos taken by orbiting satellites and space stations are certainly the most definitive proof that the earth is a sphere.
But if you're not convinced, read on...
You can also observe, with binoculars, ships slowly 'sinking' below the horizon as it sails farther and farther out to the ocean, then watch them come back. They certainly didn't fall off the edge of the earth! You can also sail or fly around the world.
The Greeks discovered that the earth is round by observing lunar eclipses (i.e. when the earth blocks the sun from the moon, casting its round shadow on the moon's surface).
Another method is simultaneously measuring the length of the shadows cast by identical poles perpendicular to a flat surface that is tangential to the earth's radius at various, distant locations. If indeed the earth is round, then the shadows should all vary in length from one distant location to another, which means that the angle at which the parallel rays of sunlight struck each pole varied from one location to another. (recall the alternate-angles theorem from Geometry class) If the earth is flat, then the lengths of all the shadows should be identical when measured simultaneously, since all rays of sunlight that strike the earth are parallel. However, they are not identical, but in fact, varies in such a way that the angles indicate a spherical surface. (This was one of the earliest methods to determine the radius of the earth)
Also, keep in mind the 24 hour time zones. When it is noon in Hawaii, it is approximately midnight in the Middle East and vice versa. How can it be noon and midnight simultaneously? It is certainly impossible with a flat earth and a sun millions of times more massive.
If I were a billionaire and physically fit, then proving to you the earth is round would be no problem. I can just take you with me on the space shuttle and we'll watch with our own eyes the earth from the orbiting International Space Station.