The thought that people who lived in Columbus' time (or any time since the third century B.C.) believed the world was flat is more urban legend than fact.
So how did this myth get started? Washington Irving’s horribly inaccurate biography of Christopher Columbus, published in 1828, and a scholarly article by French scholar Antoine-Jean Letronne, published in 1834.
Irving apparently wanted a dramatic image that would grip his readers, so he castColumbus as the bold mariner who argued courageously for truth and common sense before a council of inquisitors and hooded theologians. In fact, Columbus did appear before a council to defend his ideas. But it was a council of scientists, and the issue was not the roundness of the globe, but how far Columbus would have to sail to reach Asia. The scientists thought that Columbus had grievously underestimated the distance. And they were right.
The Greeks were among those who speculated much about the shape of the Earth. As is well known, Thales claimed that the Earth was flat whereas Anaximander made it a "stone column". The earliest claim for the round shape of the Earth known to me came from Pythagoras. It was reported that Pythagoras reasoned from the perpetual round shape of lunar eclipses that the Earth could neither be flat nor cylindrical, but only spherical. Please refer to John Burnet's Greek Philosophy Part 1: Thales to Plato. (London: MacMillan, 1914), P.44. Similar views can be found in Aristotle's opus.