You can determine the spherical nature of the Earth using your own observations -- but you do have to be perceptive.

The way the stars and shadows change with even a small shift in latitude was a dead giveaway to men ~2500 years ago. Even if you never left home, you might notice that the masts of ships on the horizon come into view before the hulls. And on second thought, why would a horizon exist on a flat world anyway?

For this example at least, you certainly could rely on first-hand evidence. You might need people to put the pieces together for you, but you can make the raw observations yourself fairly easily.