There are a ton of ways render sky and transitions. But basically boil down to different procedural generation techniques vs blending baked skies.
Though blending two skydomes works, like TheComet said; most implementations procedural generate the sky (Google "atmospheric scattering").
Since you come from an art background, I'm just going to say it's a lot of math stuff behind it, having a cool blue sky with a sun in it.
Perhaps of interest to you is that the game will usually have a set of key settings for each time of day (i.e. sun power, sun colour, and optionally different mie & rayleigh scattering parameters although that is cheating/artistic-license since technically those two params should remain unchanged regardless of time of day) and blend (interpolate) the parameters based on the current in-game time.
Atmospheric scattering doesn't cover clouds, so clouds are rendered separately: could be as a extra layer(s) of noise textures, using baked skyboxes, using oriented sprites, or using volumetric techniques.
Often clouds are rendered with a special shader to do some math to be coloured by the sun while still looking real.
In simple words, it's really technical stuff; but there is no one unique way. I was just reading yesterday slides from CryTek (starting page 108) where they were telling they tried 3 different cloud rendering techniques until they were happy (and they didn't end entirely happy).