From what I've read, Jupiter's 'Great Red Spot' is an unbelievably powerful storm. Do astronomers and scientists have any idea of what that might look like if it were possible to be inside/below it? Would it look like a storm on Earth (thunder, lightning, winds, etc.), or would it look completely different?
You would not actually be able to see that much, since sunlight there is less than 4% of its strength at Earth's distance to the Sun. That is still enough to see though (much brighter than moonlight at night in fact). The problem is the transparency of the atmosphere: we can not look in, so you could not look out. You would only see a weak brown or red glow below the cloud cover.
However, there is plenty of lightning! Strong winds and thunder exist too.
Lightning on Jupiter
There would definitely be high winds, and I assume that means very fast moving clouds. The clouds are a reddish orange, as we can tell from images of the Great Red Spot. From what I understand, it would look something like this, with lightning happening if you were in the cloud tops of the Great Red Spot, looking at the sun (granted, I could be off):
I am making this as a level for Planetary Settlers, and trying to get as accurate as I can based on what we know. Please correct me if any of the details are off. I made the sun smaller according to the size difference we would see due to the further distance Jupiter is from the sun.