Around 330 BC, Aristotle provided observational evidence for the spherical Earth, noting that travelers going south see southern constellations rise higher above the horizon. This is only possible if their horizon is at an angle to northerners’ horizon. Thus the Earth’s surface cannot be flat. He also noted that the border of the shadow of Earth on the Moon during the partial phase of a lunar eclipse is always circular, no matter how high the Moon is over the horizon. Only a sphere casts a circular shadow in every direction, whereas a circular disk casts an elliptical shadow in most directions.

The Earth’s circumference was first determined around 240 BC by Eratosthenes. Eratosthenes knew that in Syene, in Egypt, the Sun was directly overhead at the summer solstice, while he estimated that a shadow cast by the Sun at Alexandria was 1/50th of a circle. He estimated the distance from Syene to Alexandria as 5,000 stades, and estimated the Earth’s circumference was 250,000 stades and a degree was 700 stades (implying a circumference of 252,000 stades). Eratosthenes used rough estimates and round numbers, but depending on the length of the stadion, his result is within a margin of between 2% and 20% of the actual circumference, 40,008 kilometres.

Then by about 300 BC, Aristotle used simple observations to argue that the earth was round. He used the curved shape of the earth’s shadow on the moon during a lunar eclipse as an example. He also said that the fact that you could see different stars/constellations as you moved south meant that the stars were behind the horizon created by the round surface of the earth.

The first ever who got to actually experience that the earth is round is Ferdinand Magellan, a Portuguese who was commissioned by the Spanish to travel for conquest. He was able to go around the world through the Pacific Ocean to the Spice Islands, before being killed in the Philippines.

These sites are interesting and discuss the issue.