1. The overall color of daylight (direct sunlight + scattered skylight) appears white, essentially because all lifeforms adapt so that this illumination is the base. In other words, the actual spectrum of the sun is not uniform across the frequencies (a "white noise" white), but the sensitivities of the sensors in our eyes are calibrated so that the actual daylight looks neutral, to our perception. But we can also compensate for light that shifts along the orange-blue spectrum; given a brief time to accommodate, our brain normalizes the white level so that we see normally in incandescent light, which is very warm, and in pure skylight with the sun obscured, which is very cool (bluish). So it stands to reason that a creature evolved under a different sun would also perceive the normal daylight as white, the sun being a bit yellowish, and the sky being bluish (atmospheres scatter short frequency light more aggressively). So would a human astronaut, after a brief acclimatization.
2. But that is probably not what you are asking---not perceptions, but actual colors. If we assume that the sun's color does not involve magic, then it would be a function of surface temperature: the sun would emit the so-called "black-body radiation", with every point of temperature mapping to a specific spectrum. The spectrum would sum up to a pastel color ranging from flame orange to pale sky blue: to see the available gamut google black-body radiation. Colors like purple or green could not occur. This light would then Rayleigh-scatter in the atmosphere, with the result as described above: the unscattered sunlight would be redder than the actual color of the sun, and the sky would be bluer or perhaps more violet, shifted to higher frequencies. The contrast between direct sunlight and skylight would get more pronounced as the sun got near the horizon, same as it happens on Earth.
3. What could change this same-old to more exotic coloration is filtering. This could occur in several modes. (a) Magic---or sufficiently advanced tech---could filter the color of the sun before it hit the atmosphere, giving it a spectrum outside of the black-body continuum (because it would no longer be a black---read colorless---body. Duh.) The atmosphere would interact with the new spectrum, most likely again producing a situation where sky color would be shifted towards higher frequencies with respect to the sun. (b) The atmosphere itself could be the filter. This is, surprisingly, more interesting than case (a), because you no longer have the predictable bluish scattering. There are gases that are colored, like chlorine which is greenish yellow. It is toxic to most Earth life, but not necessarily to aliens. Also it could exist in the form of a layer in high atmosphere---like our own ozone layer---where its toxicity wouldn't matter, but its coloration would still change daylight, producing a lemon yellow sun in a green sky, or variations on that theme. There are other colored gases, most of them unfortunately brownish drab and quite opaque in thicker layers. But there are other ways of coloring the atmosphere, the most interesting---and frequently affecting the color of the sky even here---is particulate scattering. Most stained glass is colored this way, with tiny particles of gold, iron, or cobalt producing red, bottle-green, or blue glass respectively. The same thing can happen in the atmosphere; on Earth the most common particles are silica from deserts or salt from the sea, producing many odd colorations. The Martian atmosphere, tinted because it is full of fine dust, is a great example. Perhaps on your planet there is a constant presence of inorganic or organic particles in the air, producing just about any color shift you can think of. The one constant would be that the colors would become more intense as the sun neared the horizon, because the "filter" layer of air would be thicker; and more washed-out near high noon. With just about any of these effects, the most intense sky color would be observed shortly after sunfall.
4. And don't even get me started on clouds. Even our own, plain-vanilla water vapor clouds can produce all sorts of reflective and variegated appearances, rainbow halos, sun-illuminated cloudbanks at sunset, local darkening and brightening. Clouds containing colored chemistry, or perhaps algae-like organisms, could introduce new tints.