Does the large red shift value of galaxies mean they are far away? When the red shifts of galaxies are large, why do we think that they are  far away? 
I know about Hubble's law, Tully-Fisher relation of spiral galaxies, Faber-Jackson relation of elliptical galaxies, and so on , but they seem to the empirical rules. 
I think that the value of red shift has only the information on speed, so they might be the objects which move with the high speed near by our Galaxy.
 A: The reason why think a large red shift corresponds to large distance is that this is what is predicted by General Relativity.
If you make a few simplifying assumptions about the universe you can solve Einstein's equation for the universe to give a result called the FLRW metric. This predicts the universe is expanding and predicts the red shift increases with distance.
However we have experimental evidence for the red shift-distance relation. For example there is a type of supernova called SN1a for which we can calculate the brightness. Because we know what the brightness should be we can measure the brightness as seen from Earth and use this to calculate how far away the supernova is. Then we can measure the red shift and test the red shift-distance relationship.
The measurement of SN1a red shifts is the way dark energy was detected, because if you do the experiment you find the supernovae are actually slightly farther away than the red-shift predicts they should be.
A: The large red shift value of galaxies happens because they are going away due to the expansion of the universe.
It's happening red more and more because they're going further away more and more. Redshift and blueshift describe how light changes as objects in space (such as stars or galaxies) move closer or farther away from us. The concept is key to charting the universe's expansion.
Visible light is a spectrum of colors, which is clear to anyone who has looked at a rainbow. When an object moves away from us, the light is shifted to the red end of the spectrum, as its wavelengths get longer. If an object moves closer, the light moves to the blue end of the spectrum, as its wavelengths get shorter..
