One is talking to someone on the other side of the planet, where the time would be different (and so day or night instead of night or day).
The other - that the stars look quite different when you go to another latitude.
It may not be possible to question and personally re-check everything, but if you make that a habit, you'll notice a pattern of which experts are usually right and in which field (besides getting a good idea of how the world runs).
From there, you can bridge the gaps in your experience with carefully placed bits of trust.
You don't even need to go around the world... just call an East-coast friend from the West coast a couple of hours before sunset... "hey is it night there?" "yep" "OK, the world must be round then!"
My point was more that we take a lot for granted, some things obvious, some maybe less so, and it is not always silly to question even the simplest of assumptions.
There are everyday observations which will reinforce the fact that the Earth is not flat. Tides? Movement of Sun/Moon/Planets/Stars across the sky? Satellites? Pictures from space? That's just off the top of my head, probably hundreds more.
But, and just to play devils advocate, let's say the pictures may be fake and there is an elaborate conspiracy. Have I done any experiment myself to prove the Earth is round? There are some very simple intuitive ones mentioned here[1]. The answer is no. So while it is clear from the overwhelming body evidence that it is not flat, I have accepted almost all of that evidence without verifying it myself in any experimental way. And this is one of the simplest topics one can examine.
I haven't tracked the movement of stars across the sky, I have not been to space, I have not measured shadows or the curvature of the earth. I've been in a plane, and it does appear to be curved, so there is that.
Obviously, I cannot possibly question everything, it would be futile and exhaustive and get me nowhere. But I do believe we do not question enough the truths we take for granted because they have been presented to as truth.
And while some topics are barmy, flat-earth being one, there may be others with much shakier body of evidence to back it up, which are today presented and taught as truth. Also, I find myself more and more intrigued by a good conspiracy, and enjoy the company of conspiracy theorists, because it makes me think and re-examine why I take the presented truth for granted.
[1] http://physics.stackexchange.com/questions/26427/what-is-the...