I know that the educated people knew about the spherical Earth because of works left behind by Greek and Roman scholars. However, laypeople back in the day had no access to such knowledge, so what did they believe about the Earth's shape?
This is a tricky question to answer, because by definition if someone leaves behind a written record where they're casually referring to earth as a sphere, they are educated (as in, they know how to write). 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. I wrote about that some here.
A fairly accessible source on this (easy to read and any university library should have it) is Ronald H. Fritze's New Worlds: The Great Voyages of Discovery, 1400-1600.