Imagine you are a common man and want to prove that the Earth is Round, how would you prove it without any mathematical derivation or without the theory of the ships.
Just look at how people actually did it. ;) Measuring shadow length at the same time at different places, comparing horizons (as also suggested in comments),... Also have a look at this list, some of which are down to earth methods. ;)
By the way, there are easy ways to prove that the earth is rotating, like the Foucault Pendulum, as well.
The fact that you can see the horizon is an indication that the Earth is round. If the earth was flat on a clear day if you chose the right spot you should be able to see to the end of the Earth (or at least the nearest mountain range). From memory the distance you can see from ~1m above see level (think sitting in a lifeboat) is ~5km due to the curvature of the Earth.
For many reasons this is a somewhat unsatisfying logic. You could easily argue this effect is due to haze/atmospherics and you rarely have unobstructed views anyway.
A more convincing argument used by the ancient Greeks is to consider shadows case by a pole (or down a well if you want to be historically accurate). Near the equator on the summer solstice at noon the sun will be directly overhead and cast no shadow. At the same time at another town some distance away the sun still casts a shadow. If we assume that the sun is far away so the light beams are parallel then we can conclude the Earth must be curved. Furthermore if you are prepared to do some maths using the angles of the shadow and the distance between the two points you can calculate the circumference and diameter of the Earth as was done by Eratosthenes around 240 BC.
To prove that the Earth is round, look at sunset on clear days. The sky is bright after the Sun has gone below the horizon. If there are some clouds these are still lit by the sun and cast crisp shadows. If the Earth were flat, the sky and clouds would get dark at the same time and any shadows would grow more and more feint.