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The ancient Greeks (starting at least with Pythagoras, 6 centuries before Christ) had worked out the earth had to be a sphere (using scientific principles), by the time of Pliny (1st century AD) the notion was generally accepted, including by most of the church fathers, such as Augustine, and many later clerics (such as the Venerable Bede).

Credit for a convincing argument that the earth is a sphere is generally given to Aristotle. He used the usual arguments (observations of mariners, etc.) plus the fact that the earth's shadow on the moon during lunar eclipses is always circular, and only a sphere could always cast a circular shadow.

A generation later, Eratosthenes provided a strikingly accurate calculation for the earth's size, by measuring the shadow of two objects at noon, which were a known distance apart (the figure of 500 stadia is often given). The sun was directly overhead for one object (a well which was lit up right to the bottom) and about 7 degrees off-center for the other, so every 7 degrees was 500 stadia, and for 360 degrees ... he had the earth's circumference. (His writings are lost, and there's another way he might have done it, but this post is long enough.)

This was around 250BC, and all educated people from then on knew not only the shape but also the size of the earth. Columbus, however, used a different calculation, and didn't realize how far he had to sail to get to China. But he knew the earth was a sphere.