Here is the only needed evidence to prove that the earth is round:
1) Gravity exists | We can observer the gravity exists from the tides rising when the moon or sun is overhead. The gravity from the sun or moon will exert its gravitational pull on the water, causing the level of the water to rise.
2) Gravity intends to pull everything into a sphere like object (a spheroid) | Just look at any celestial body (eg. the moon). Any body with enough mass will intend to pull itself into a spheroid. The moon, sun, and every other planet is a spheroid because of their gravitational pull.
3) The earth is massive enough to be classified to be a planet | We can observe the affect of the earth's gravity on the moon, and you can calculate that the earth is roughly 6 time larger than the moon. If the moon has enough gravity to pull itself into a sphere, then the earth should have more than enough gravity to pull itself into a sphere.
In conclusion, because gravity exists and gravity will always attempt to pull a massive object (like the earth) into spheroid (just like every other object in the solar system), the earth is sphere shaped.