How does red shift (Galaxies moving away at various speeds) tell that the universe is expanding?
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Red shift in light is similar (although not exactly the same as) the dopplar effect. The doppler effect is seen when an object which produces waves (sound, light, or even physical waves) moves through a medium, the waves are stretched or compressed when observed. This can easily be seen when looking at the bow of a boat or listening to a siren on a vehicle driving by at high speed. The waves in front of the object are compressed, while the ones behind the object are stretched.
Since light moves as a wave, when the source of the light is moving (or in this case, the space between the sources is changing size), the waves appear to the observer to be compressed (more blue) or stretched (more red).
Red-shift happens when the object being observed is becoming more distant. (I use this awkward phrasing because in this case the galaxies in question are not moving away, but rather the space between them is growing)
If we observed galaxies on one side of us with a red-shift, and ones on the other with a blue-shift, that would indicate that we are the ones moving through space.
However, this is not what we observe. What we actually find is that almost everything we look at has a red-shift, meaning that they are all getting further away. This means that instead of us moving, everything around us is expanding.
One thing to note is that this does not mean we are in the center of the expansion, but rather that space is expanding uniformly. This means that if we were to observe the universe from a different location, we should see the same effect. This detail is still under some debate btw, with some studies finding slight variances in the direction of expansion.
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