Pythagoras postulated that the Earth is round in about 500BCE. However there's a line in Homer's Odyssey which suggests that seafarers had known the Earth was a globe for at least two or three centuries prior to that.
Plato (around 400BCE) not only assumed that it was round (as did nearly all Greek scholars) but put forward an estimate for its circumference.
His student, Aristotle, went a long way to proving that the Earth is round
Archimedes, around 250BCE, improved upon Plato's estimate for the size of the Earth.
Eratosthenes (around 200BCE) put forward a theoretical experiment to find the Earth's size. If it had actually been carried out the result would have had an error of less than a half of one per cent.
Indian scholars were able to calculate the size of the Earth to even greater accuracy by around 500CE.
Islamic astronomy assumed from its outset that the Earth is round (based upon what their scholars had learnt from the Greeks).
Even in the Middle Ages (when history books have often taught that most people believed in a flat Earth) nearly all scholars accepted that the Earth was round. The idea that the Catholic Church insisted that the Earth was flat seems to have derived from the falsehoods put about by anti-Catholic Protestant propaganda in the 17th century and has nothing to do with reality.