Our eyes are adjusted to the sunlight in daytime, which means that the sun itself during the day appears white. Light with higher frequencies appears blue, light with lower frequencies appears red. Once a white light goes through the atmosphere, blue light scatters more, so the atmosphere -- that is the piece of sky around the sun -- appears blue. (it scatters in all directions, but some of the scattered blue light goes into our eyes)
When we talk about evenings, the light has to pass through a thicker layer of the atmosphere. This means that more of blue light scatters on the way from the sun to our eyes (some part still scatters into our eyes, but some is scattered into other directions). The more blue light scatters the less goes trough. This is why, comparing to daytime, the sun itself appears more red. Still, the sky around the sun should be somewhat bluer then the sun, as during the day. Roughly, the light that scatters in upper layers of the atmosphere is primarily blue. But to reach you eyes it goes trough other layers and scatters there, so when it reaches you a large portion of blue is scattered and it becomes more red. The light that scatters in lower layers is already red, as only red light from the sun reaches there. So, eventually, the sky is also red. Of course, to make it accurate, you should rather solve a differential equation considering scattering at each infinitesimal layer of the atmosphere.
The key thing that needs to be highlighted here is the following. The color of the sun is defined by the dominating frequency of the light emitted by the sun minus the light scattered by the atmosphere. Instead, the color of the sky is defined by the dominating frequency of the sunlight which is scattered by the atmosphere. So, these colors are formed in a sort of complementary fashion.