...well...yes and no. For basically all of our beliefs about things outside of our immediate environment, we rely on evidence that other people have collected. I can't determine that the earth is a sphere by my own observations any more than a man could 5 thousand years ago.
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.
If the radius of the Earth is r, and you have height h, then from the top of your head to the center of the Earth to the horizon and back to the top of your head is a right angle triangle. The hypotenuse is r+h, one side is r, and the other must be sqrt(2rh + h*h). Under the assumption that r is much bigger than h, that means that the distance to the horizon varies as the square root of the height.
This assertion is surprisingly easy to check with ships. Furthermore if you pay close attention on a commercial aircraft, you can actually see that the horizon is a little bit below level!