How to quickly or easily prove the world is spherical?
To answer my own question based on the useful comments here, my favorite methods were:


The earth's shadow cast on the moon during an eclipse. It is easy to observe and the flat earth counter argument is so insane that if it is offered you should just walk away.
Shadows differ from place to place. The same experiment as Eratosthenes but you can carry this out locally with a piece of large card, 2 obelisks and a light source casting a shadow. Now bend the card and observe the change in shadows. The same effect could then be observed on the earth's surface. The counter argument of curved rays of light is weakened by the fact you can reproduce it locally.
triple-right triangle:
You move in a straight line for a long enough distance Turn right
90° degrees, walk in that same direction for the same distance Turn
again to the right 90° degrees and walk again the same distance
After this you'll end up at the starting point. This is not possible
on a flat surface since you'd just be "drawing" a half-finished
square. Not sure how practical this is as you'd have to walk a long way!
Although not technically about the earth being flat, in a flat earth theory the sun orbits, or at least moves over the earth. A simple way to discount this is observing parallax with surrounding stars.
It is not too difficult to prove the earth curves at least, such as by observing more objects over the horizon the further you go up. And even measuring the curvature is not difficult. Thus I believe a person could drive the length of the African continent, observing the curvature and measuring the straight line distance, and modelling that alone would show the earth curves beyond what could be acceptable on a flat earth, without all the water running off the edge!
One I haven't seen mentioned yet is showing that the flat earth map cannot work. In a flat earth, apparently, the north pole is at the centre and south pole is a ring around the outside. Pick a land mass close to the south pole, and another on the opposite side of the map. The distance you would need to travel to reach one from the other by e.g. flying east around the map would be enormous and not congruent with the flight times or shipping times in real life.
Earth is warmer near the equator, for obvious reasons in a non-flat world but the counter argument would have to be pretty imaginative.
Sunrise and sunset do not happen at the same time for everyone on earth. The difference in timezones, which can be observed by taking a trip to different countries, cannot reasonably be explained by a curved but nearly flat earth.
Experiments with gravity, but flat-earthers reject the complication of gravity (things drop just because). So I avoided this one, but it would be a compelling way to demonstrate why people on the other side of a spherical earth are not on their heads.