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.
It is a fun exercise to figure out how far away the horizon is.
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!
Adding in that the Earth's radius is about 6.4 megameters, and starting with h=5 meters you get that the horizon is about 8000 metres away. So from 5 meters up the horizon is about 5 statute miles away. Then to double the distance, multiply the height by 4.