Humans have known for at least 2300 years that Earth is "round"; although most thought it was a sphere (close enough to the truth), there were, back then, some people proposing that it might be cylindrical. Eratosthenes calculated the circumference to within 10% of the actual value. Not only they knew it was round, they even had a good idea of how big.
The idea that people thought the Earth to be flat, hence giving Columbus all kinds of problems to have his project approved and financed, was invented by an American author (Washington Irving) in the 19th century: A History of the Life and Voyages of Christopher Columbus (1828) which was meant to be a biography of Columbus, but is what polite modern historians call "Romantic History" (and the rest of us call "fiction"). Unfortunately, president Tyler made Irving the Minister to Spain, giving his book far too much credence: it was used as a school manual for a few generations, thus ruining the perception of Earth (and humanity) for a few generations of Americans.
As for the "Biblical" flatness of the world, it is important to know that in the original versions of the Scriptures (before the Bible was created by translating and compiling separate stories), the proto-Hebrew word used for "world" is defined, in Genesis, as a region surrounded by 4 bodies of water (turns out to be two rivers, one lake and one sea): the borders of what was, back then, Israel. Certainly, parts of THAT "world" appear flat, compared to the rest of the planet...
On Earth's surface, any sufficiently large triangle, even when allowing for corrections due to differences in altitude, will show "spherical excess" (the three angles add up to more than 180 degrees) that is proportional to the area of the triangle (twice as large = twice the amount of excess), regardless of where you put the triangle, thus proving that Earth's surface has positive curvature everywhere (therefore, it cannot be flat).