This is a fairly common question in this subreddit. I've written about it a few times, as have others:
To quote from that last answer,
Hipparchus of Nicaea (c. ~190-120 BCE) seems to have been the first person to propose using a grid system to find the position of cities (and other places) on a globe, which implies an understanding of longitude. He built on earlier work by Eratosthenes of Cyrene (c. 276-194 BCE), who had mapped the known Earth, including finding its circumference. Hipparchus' method of finding the longitude of places was to use the differences in timing of lunar eclipses at different points on the globe to calculate the difference between local time of those points; the drawback is that there was no accurate-enough method of timekeeping to lead to useful calculations. Edit to clarify: The difference in local time between observed beginning and end of the eclipses would serve, essentially, as a way to understand the longitude between places.
and from the first one:
But we can be pretty sure that a spherical Earth was commonly known from a couple of sources:
the Elucidarium written by Honorius Augustodunensis around 1120 refers to the Earth as a sphere. The Elucidarium was a manual for the instruction of clergy, and was translated into multiple languages. (I'm copy-pasting these from Wiki because I don't feel like retyping them: "Middle English, Old French, Middle High German, Old Russian, Middle Dutch, Old Norse, Icelandic, Spanish, and several Italian dialects.") The lesser clergy were those most likely to preach and minister directly to commoners, of course.
We have surviving sermons from Bertold von Regensburg, a German cleric, that refer to the Earth as spherical (not in the sense that he's revealing a great mystery of the Earth, but as an analogy that his hearers would be familiar with). His sermons were in the vernacular German dialect of his time period (mid-13th century), so there's evidence that he was using that metaphor in the local language that the local people would be aware of.
The question that early explorers faced, of course, wasn't so much whether the world was a sphere as it was how big it was -- Columbus in particular overestimated the size of Eurasia and underestimated the Earth's circumference.
Now, that said, those answers above are very Eurocentric and I'd be glad to hear from more people on the question of how other cultures thought of the spherical Earth.