We say Light is Red-Shifted or Blue-shifted from faraway stars and galaxies We say Light is Red-Shifted or Blue-shifted from faraway stars and galaxies. Can we find out the distance at which it changed its frequency.
So in another solar system, it might seem to be Green shifted or something similar. (Assuming that those beings can see only in our visible region)
And why do stars still appear white in the sky? If the rest of the universe is going away from us.
Is it only a misconception or misunderstanding. Please let me know. 
 A: There is no such thing as "greenshift"
First off, the terms "blueshift" and "redshift" correspond to light getting shorter or longer wavelengths, respectively. We use these terms because blue light has a shorter wavelength than red light, and we use them irrespective of the actual wavelength of the light. That is, if an ultraviolet photon with a wavelength of $\lambda=1216$ Å travels through the Universe until its wavelength has increased by a factor of 3.3, it will now be in the blue region of the visible spectrum, but we still say that it has been redshifted.
Redshift is a continuous process
Second, we can't say at which distance it changed its frequency, since this happens at all distances. The reason is that the redshift is a result of the expansion of space, and space has been expanding at all times since the Big Bang. However, in principle you could imagine a universe that was static when the light from a distant galaxy was emitted, then as some point in time expanded violently by a factor of, say, 2.3, and then again was static. In this case, when you observed the light, its wavelength would again be stretched by a factor of $(1+2.3) = 3.3$, and you would still see the aforementioned light as blue. And you could then say "its redshift happened at a distance of $[\ldots]$".
The inverse square law $\ne$ redshift
The flux we receive from a nearby light source decreases with the distance squared. This is known as the inverse square law. The same is true for a distant galaxy, but in addition, the flux decreases due to the light's energy decreasing with the redshift. These are two unrelated effects.
Stars are white because they are dim, nearby, and Solar-like
All the individual stars you see with the naked eye are inside our Milky Way. Although they're moving with respect to you, their motion are not (on average) away from you, and thus they're not redshifted. Also, their motions are way too small to notice any red-/blueshift with the eye.
Additionally, the typical star emits, like our Sun, most of its light in the wavelength region that human perceive as "white". As chaz327 say, hot stars emit more blue light, and red stars emit more red light, and if you're standing in a dark enough place, you can actually see this. For instance, the constellation Orion has a very red left shoulder (Betelgeuze, temperature ~3500 K), and a bluish right knee (Rigel, temperature 12,000 K).
However, the human eye has a hard time discerning colors of very dim objects. We see colors with the cone photoreceptors in our eyes' retina, but these aren't very light sensitive. So in low-light conditions, we use the rods instead, which can't detect colors but only light/not-light.
A: It is my understanding that virtually all galaxies are moving away from us, and so red shifted, except for a few galaxies (including dwarfs) in the local group, like Andromeda, which would be blue shifted.  So, most galaxies are red shifted, and the further away they are, the more red shifted they are.  So the the change in frequency is relative to the distance the galaxy is from us.
The current theory(?) is that no matter what solor system you are in, you would see the same effect, that most of the galaxies are moving away from you, and the further away they are, the faster they are moving away, and thus red shifted.
Stars emit all colors of light, so they appear to be white.  Small stars emit much more red light, the sun emits more yellow light, and large stars emit more blue light.  I could be wrong here, but I'm pretty sure it's spectroscopy that helps us determine a star's 'color,' not what we see with the naked eye. 
https://en.wikipedia.org/wiki/Spectroscopy
https://www.youtube.com/watch?v=hI8gifMbeVU
https://www.youtube.com/watch?v=Dq7fmftneaY
the last two videos should answer most of your questions
