Simple check for the global shape of the Earth

I love this question, because it's a very simple demonstration of how to do science. While it's true that in science one should never accept anything 'without evidence', it's also true that blind skepticism of everything and anything gets one nowhere - skepticism has to be combined with rational inquiry. Your date has gotten the 'skepticism' part of science, but she's failed to grasp the equally-crucial part where one looks at the evidence and thinks about what the evidence implies. You cannot just refuse to think or accept evidence. If your goal is to learn nothing, then nothing is what you'll learn.

There are many, many ways of verifying that the Earth is not flat, and most of them are easy to think about and verify. You certainly do not need to go to space to realize the Earth is round!

If the Earth is flat, why can't you see Mt. Kilimanjaro from your house?

Mt. Kilimanjaro is tall, probably taller than anything in your immediate neighborhood (unless you live in a very deep valley) and so the question is why wouldn't you be able to see it from anywhere on Earth? Or, for that matter, why can't you see it from even closer? You have to be really close, in planetary terms, to be able to see it. This wouldn't be true if the Earth were flat!

One might argue that this is just because of the scattering of the atmosphere. Distant objects appear paler, so probably after some distance you can't see anything at all.

So then let's think about things that are closer. Stand on the ground and the horizon appears only a few km away. Go to the top of a hill, or a large tower, and suddenly you can see things much farther away. Why is this the case if the Earth is flat? Why would your height above ground have anything to do with it? If I raise or lower my eyes with respect to a flat table, I can still see everything on that table. The 'horizon' of the table never appears closer.

If the Earth is flat, why do time zones exist?

Hopefully your date realizes that time zones exist. If not, it's pretty easy to verify by doing a video call with someone in a distant location. The reason for time zones, of course, is that the sun sets and rises at different times at different parts of the globe. Why would this be the case? On a flat Earth, the sun would rise and set at the same time everywhere.

If the Earth is flat, why is the Moon round?

The moon is round and not a flat disc, as you can see by the librations of the moon. What makes the Earth special, then?

Further, all the planets are round, although to verify this you need a good telescope. Again, what makes the Earth special?

If the Earth is flat, then what is on its 'underside'?

Hanging dirt and leaves? A large tree? Turtles? Those who reject the roundness of Earth either have no explanation or their explanation is based on much less solid grounding than the pro-round arguments (which, of course, is because the Earth is not flat).

If there is 'nothing' under the Earth, then lunar eclipses would make no sense as the Earth needs to be between the Moon and the Sun.

EDIT: As to the question of whether the Earth is round or some weird hemisphere/pear/donut shape, among other things those would all lead to a situation where gravity is wrong. For a hemisphere for example, gravity would not point down (towards the Earth) at any point on the Earth's surface unless if you were sitting right at the top of the hemisphere. Similar arguments can be made for the other shapes.

Sure, it's possible to make it 'work' by doing even stranger things like altering the distribution of mass and so on, but at that point you've gone very far into violating Occam's razor.

No matter which evidence you bring up, it can — and routinely will be by a Flat-Earther — refuted using a counter-argument along the lines of "You believe so because that's what you were taught". Unless you want to derive all of geometry, carry out all physical experiments through the history of mankind, etc., in front of the FE'er, every argument you can bring up ultimately hinges on your faith in other people's mathematical derivations and physical experiments, which, in the eyes of the FE'er, are flawed and/or deliberately altered.

Take for instance all the arguments given in the other excellent answers:

• The Earth's shadow on the Moon during a lunar eclipse would, at least sometimes, show any non-roundness of the Earth
  • Lunar eclipses are caused by a satellite called "the Shadow Object".
• If the Earth is flat, why can't you see Mt. Kilimanjaro from your house? (and many similar arguments like "Why do you see the mast of a ship before its hull?", and even "Why isn't it day all the time?")
  • Even a on flat plane, parallel lines do not extent to infinity. What is even infinity? Infinity doesn't exist. The effect is caused by the angular limits of perception.
• If the Earth is flat, why do time zones exist?
  • They match the position of the Sun's circular path above the disk.
• If the Earth is flat, why is the Moon round?
  • The Moon is a planet, like Mars, Jupiter, etc. The Earth is not a planet.
• If the Earth is flat, then what is on its 'underside'?
  • I don't know, and neither do you.
• A camera attached to a balloon shows the curvature of Earth
  • The curvature is caused by a fish-eye effect. Or just photoshopped.
• Stars travel opposite ways on the northern and southern hemispheres
  • The stars move in mysterious ways.
• If the Sun is so close to us, it should have a smaller angular diameter, the farther away it moves from us
  • Magnification caused by the intense rays of light passing through the strata of the atmolayer (yup).

The reason of this is, in my experience, that FE'ers fundamentally do not understand how science works. They don't distinguish between laws, theories, hypotheses, facts, and ideas, they refuse to acknowledge helpful concepts such as Occam's razor and Russell's teapot, they cherry pick whatever experiment remotely seems to confirm their idea, and they disregard all evidence not confirming their idea as flawed/faked.

I realized this last year, when I participated in a radio program together with a FE'er. We went to the beach with his $87\times$ zoom camera, in an attempt to see the shore 19 km away. If he were right, we should be able to see the shore, and if I were right, we shouldn't be able to see the lower roughly 15 meters of the buildings there, disregarding atmospheric refraction. I was prepared that in fact we should probably see more of the buildings due to refraction (and was a little afraid that we would actually see the shore, since this was in the spring where refraction in warm air over cold water may be higher). It turned out we couldn't see the lower roughly 7 meters. Rather than taking this as evidence that at least the Earth isn't flat, he held it against me that my theory obviously is wrong. When I asked for at least some mechanism that might produce a flat Earth, he merely claimed that I also couldn't explain what might produce a round Earth. When I mentioned "gravity", he virtually quoted Phoebe from Friends: "Don't get me started on gravity".

I think the best you can do, if your date/friend/skeptic cousin is not already a completely convinced FE'er, to say that,

If indeed the Earth is flat, then either

1. All physicists of the world are in some evil plot against humanity.
   • This would imply that, whoever is behind this, would have means to make us all keep our mouths shut. There are roughly $\sim10^6$ physicists in the world today (Day 2015), and including past physicists increases this number to several millions.
2. All physicists are incompetent.
   • This would imply that, so far only pure luck has held up airplanes in the sky, made communication possible through small, hand-held devices, enabled visually impaired people to see normally through metal-embraced pieces of glass in front of their eyes, etc. etc.

There is no proof that neither of the above is the case. But you could turn it around and claim that either 1) all members of the Flat Earth Society know that the Earth is round, but have some secret reason to keep this knowledge hidden from the public (implying that all $\sim10^3$ members$^\dagger$ of the FES keep their mouth shut), or 2) all members of the FES are incompetent (implying that… well, I don't really know what this implies).

How to prove to yourself that Earth is globally round

As in all physics, there is no proof. Physics doesn't prove anything, it verifies hypotheses continuously through experiments — preferably in several, mutually independent ways — until at some point we believe that a hypothesis describes something true, at which point we may call it a theory (or a part of a greater theory).

There are many ways to verify that Earth is round, both locally and globally, but I think they all require you to either believe that other people are not a part of a plot (e.g. you'd have to believe that pictures of Earth taken from space are not faked), or to travel a bit (e.g. go to several locations and look at ships/bridges/land disappear below the horizon).

I don't see a way that, staying in the same place and only relying on your own eyes, you can check that the Earth is globally round. But if you will allow to contact friends in different locations, you can of course compare, e.g. which way stellar constellations are rotated, or the angle between the Sun and the horizon.

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$^\dagger$_{I couldn't find any present numbers, but The Flat Earth Society had 189 members in 2010, according the their vice president.}

Look at the Moon during a lunar eclipse. Or during any other Moon phase for that matter, where part of the Moon is shaded.

As ironically exemplified by Neil deGrasse Tyson in a Twitter tweet:

Neil deGrasse Tyson @neiltyson
^{November 26, 2017}

A Lunar Eclipse flat-Earther's have never seen.

We always only see a circular shade as if the Earth really is a sphere. Sure, it is possible even with a flat Earth that the arrangement of Sun-Earth-Moon just happens to give a circular shade by coincidence. But then we would only sometimes see a circular shade - other times we should see something like the above image.

Just wait for the next lunar ecplise or Moon phase. Or for the next 100 ones. Surely, at some point you must be seeing the shade from another angle giving an elliptic shade. Or a thin shape as the picture.

These observations can be done with our own naked eyes. And we have never, ever observed any sign of a flat-Earth shadow.