There are actually quite some people out there that rejects the paradigm that the earth is a spinning globe doing orbits around the sun, which again is doing orbits around the galaxy - ref 
I also stumbled upon the flat earth society wiki a useful resource for anyone that wants to explore this alternative paradigm.
If I understand it right, in the flat earth paradigm there is a massive conspiracy going on to hide the fact that the earth is flat. Under that paradigm, you basically can't trust things you read on the Internet - hence I will try to focus on the things I've personally observed and what you also can observe - plus some observations done by people I trust. Still, I can't help it, I also have to dwell a bit on how deep such a conspiracy has to run.
Navigation
Image source: Biblioteca General Antonio Machado
I'm a sailor, unfortunately I don't have a sextant in my toolbox - but I do know the principles. With a decent timepiece and observing the angles to the stars one can find the longtitude - and by observing the angle i.e. to the polar star even without consulting a clock, one can find the latitude. Sailors in earlier times were pretty much dependent on the sextant and good timekeeping. Even without actively using the sextant, it's easily observed by me - I've moved some 1600 km south and indeed, the polar star is visibly lower on the sky here than in my former home town. Most of us should be aware of how this works out in the round earth paradigm. For those interested in how it's explained in the alternative paradigm, I'd recommend visiting the wiki.
On a related note, I've even visited countries much further south, and observed that the sun was almost right above my head - while in the north the sun is pretty low on the sky even in the middle of the day in the middle of the summer. Those southern countries experiences a rapid transition from day to night - while in the north there are often several hours of twilight. This is also covered by the round earth theory - one can relatively easily explore it by using a lamp and a globe. Now I'm staying north of the equator and are using the words "north" and "south" - I do suppose there are flat-earth-theories explaining this pretty well, but I also suppose those flat-earth-theories will break on the southern hemisphere. I've never been there, but there are lots of people that has experienced the polar day and polar night at the Amundsen-Scott South Pole Station.
Today we've abandoned the sextant, and we're using GPS instead. In the round earth paradigm, the GPS'es works by tracking multiple satellites and checking the time it takes for signals to reach from the satellite down to the receiver. As a bonus, the receiver also gets very accurate timing information. I'm wondering how the GPS technology works in the flat earth paradigm? Under the flat earth paradigm, there must be million of engineers out there that pretends the GPS units work by tracking satellites, while they secretly know there is some other magic/tech at play. And yes, I can personally attest that the GPS works pretty well.
Midnight sun
In my former home town we had like two months with the sun being below the horizon in the winter time, and two months with the sun above the horizon in the summer time.
Again, anyone can play a bit with a lamp and a globe and understand this using the round world theory.
Day and night
People were observing this in the pre-telecommunication era, as well, but today it's super-easy to check this out: while it's day at one side of the globe, it's night at the other side. Anyone can run up a video conference, even with some random dude found on the Internet. Oh, hi Peter, I hope I didn't wake you up?
Again, anyone with a globe and a flashlight can see how the round-earth-theory explains this, and astronomers can with great certainty predict exactly when the sun will rise and set on different latitudes. Also, one doesn't need to be an astronomer to be able to predict roughly when the sun will rise, set and be in the south at any location of the earth.
Circumnavigation
I do know several persons that have traveled around the globe and come back to the original point. Ok, the flat-earthers do have a point: it's also perfectly possible to travel in a circle on a flat map and come back to the original point. However, according to their world map it should be a very long trip to travel around the south pole. Well, again, they may have a point, sailing around the Antarctica is a big and dangerous trip indeed. In their paradigm it's impossible to cross the south pole, so a circumnavigation over the poles is pretty much impossible in this paradigm.
Quite some Norwegians have been exploring Antarctica, but I must confess I don't know anyone personally that have been doing a circumnavigation over the poles.
Sunsets and sunrises
Where does the sun go during the night? In the flat earth paradigm, if I understand it correctly, the sun is simply too far away to be seen during the night. I've seen quite some sunrisings and sunsets during my life - and what I see corresponds very well with the round-earth theory, but not at all with the flat-earth theory.
Solar and lunar eclipses
Image source: Wellcome Trust
"Round-earth" scientists are today able to predict with great accuracy not only when and where the next eclipse will be, but also exactly were on the globe it will be visible. I've witnessed this myself, in my youth I did an eclipse trip and ensured I was at the right time and location. I was to meet my (future) wife there, but for various reasons she missed it and didn't see the total eclipse.
If I understand the flat earth theory correctly, said scientists are all participating in some conspiracy, they do their calculations using the flat earth theory, but presents it as round-earth-science? I guess I'm just not sufficiently conspiracy-minded to believe in such things.
The "sunken ship"-effect
Out on the open sea, a boat in the distance will appear to be "half-sunken", i.e. only the mast visible. Weirdly enough, I can only once clearly remember to have observed this effect - arriving to Denmark by a big ferry - Denmark is a quite flat, so in the distance the big houses looked like if they were growing directly up from the sea.
Of course, it could be some optical illusion, and the flat earth society has an explanation to this phenomena as well.
Satellites
Ok, I've never been much interested in satellite TV, so I can't personally attest that one gets a signal if pointing the satellite dish towards a geostationary satellite, but not if pointing it somewhere else - and at my longtitudes one has to point the dish quite flat, so in theory the sender could be at the top of a mountain rather than from a satellite. Still, there are hundreds of millions of people in this world that has been pointing their satellite dishes up towards the sky to get TV signals.
My hometown in the far north was a good place to receive signals from satellites in Polar Orbit, of course I knew people working with satellites. I've also seen artificial satellites passing by on the sky. I think some weeks ago there was an announcement that the ISS would pass the moon or something like that - anyone wanting to check that the ISS exists, is flying over our heads and that we have a pretty good overview of where it is could just point their binocular towards the moon at that point. Sadly, I didn't.
I've also seen satellite photos. Amazingly, that half a decade ago American U2 pilots risked their life searching for the Baikonur Cosmodrome. Some years ago I had some idle time and was playing a bit with Google Earth - and incidentally found the Cosmodrome while doing so. According to the flat-earthers such satellite photos must be fake.
There are millions upon millions of people directly or indirectly working with satellites, according to the flat-earthers they must all be part of a big conspiracy.
Weather and tides
Wind, waves, tides and the golf stream current ... I have personally experienced all of those. I also know there exists trade winds making it easy to sail in both directions across the Atlantic and Pacific. Admittedly I haven't personally crossed the Atlantic nor the Pacific, but at least I've experienced that Denmark very often gets wind from the west and rarely from the east. Those things are easily explained through the Coriolis effect.
Last words
Consider quantum mechanics for a while - the theory flies in the face with what most people would consider sane - but it's still a very useful tool, it can be used for predictions.
Lets for a moment consider that the earth is round, but that it also is the center of the universe. Everything is relative, I don't mind that you choose to define the earth as the center of the universe, and I can't prove you wrong - but the mathematics describing the planet orbits etc in such a reference frame is extremely complex. If you're going to send a space probe to Jupiter, it's very much easier to do so based on a reference frame where the sun is the center of the solar system.
Lets for a moment consider that the earth really is flat, and that there exists alternative explanations on how all the above and more works, not involving the earth being a spinning globe orbiting the sun. Of course, anyone can see that the earth is flat, the round earth do indeed fly in the face for quite some people. I'll tell you what: it does not really matter! Even if the earth happens to be flat, the round-earth theory is a very useful toolbox, it gives useful predictions, it has helped people navigating on sea for many centuries, it still aids people navigating by makes GPS'es work, it brings TV-signals to peoples home, and the theory is even relatively easy to grasp and understand for non-physicists.
I may of course be very much biased through decades of indoctrination, but I find the flat-earth theories (as they are presented on the flat earth society wiki) to be ... rather complicated and twisted, and I seriously doubt that it's possible to make useful predictions through those theories.
The wiki lists some evidences that the earth is flat as a pancake, to be honest I haven't had the time to look through them. That's the case with any scientific theory, there will always be "evidences" against them - for instance, Galileo predicted that a big and a small iron ball would fall to the ground with the same speed and demonstrated it - but exchange the small iron ball with a feather, and obviously it falls much slower. Sometimes one can find a reasonable explanation, like, the reason the feather flies slower is due to air resistance (indeed, in Apollo 15 one of the astronauts did this experiment again with a feather pen, and demonstrated it fell with the same speed as a hammer on the moon), other times eventually the old theory gets scrapped and replaced with a new one (like, Newtons laws are superceded by Einsteins laws - but "Newtonian physics" are still very useful). I think the "evidences" against the round globe theory are very weak compared to the mass of evidences against the flat earth theory.
Conclution
If you seriously believe the earth is flat, I'm sorry to say that I seriously believe you're bonkers.
Postscriptum
I vouch that I will upvote any comment here rationally trying to argue for the flat earth theory, even if I think you may have gone bonkers.