How do we prove the earth is round? What evidence do we have and reasoning?

Just like any other scientific conclusion ("theory"), we DON'T say the conclusion first and then go out and look for evidence that supports it. The "scientific" process is to gather up ALL the evidence we possibly can, and THEN figure out a simple description that can explain the evidence. After that, if our description turns out to predict things that we HAVEN'T seen yet but they do turn out to be true, that gives us more confidence that our description is good, and probably true and accurate.

When you live for several years, and you talk to a lot of people, and there's a lot of discussion of day and night, sunrise and sunset, moonrise and moonset, the looks and motions of all the things in the sky seen from different places at different times, how mountains and clouds look during twilight, how ships look when they're coming in and going out, how distant mountains appear, the idea of the Earth as a spinning sphere is the one that can explain them all. No other description can, unless it's hilariously complicated to start with, and you keep making up new kinks and patches every time somebody finds a problem with it.

This is where we were about 2000 years ago.

Since then, telescopes were invented, and we could see that all the OTHER planets are spinning spheres.

And just yesterday, when we started flinging cameras out, turning them around, and taking far-out pictures of the Earth, guess what ! The pictures show that the Earth is spherical. And when we take videos, we see it spinning.

Our theory is looking better all the time.

. west