If our solar system and galaxy are moving why do we not see differences in speed of light depending on direction? May be a silly and simple question, but I've been wondering if:
The speed of light is constant, and

*

*When we're moving in the same direction (where both the emitter and the receiver move with the light direction) we would be making it take more time for the light to reach the other end.

*Conversely when moving in the opposite direction we'd be shortening the time it takes.

Why do we not see a non-uniform speed of light caused by solar/galactic movement?
Or do we?
 A: Judging by your comments, the point you are missing is that the speed of light is constant relative to everything. Suppose you have a friend who is standing still, and you pass her at half the speed of light at the same instant that I overtake you at three quarters of the speed of light, a light ray will pass all of us at the speed c. That means your stationary friend will consider that the light is moving away from her at a speed c, you will think it is moving away from you at a speed c, and I will think it is moving away from me at a speed c.
A: It is not a silly question at all, and posed a problem until Einstein explained the results of Michelson–Morley experiment using special relativity in 1905. This experiment sought to measure this exact observed difference in the speed of light that you would imagine would result from the Earth's motion. They found that it remained invariant, and Einstein resolved this discrepancy by saying that the speed of light must be held constant for any observer. It’s not the easiest thing to imagine, but it has yet to be disproven and it makes many useful predictions.
What does happen as a result of relative movement is called a Doppler shift. Similar to classical sound waves, the observed frequency increases as you move towards a source, and decreases as you move away. Instead of this shift corresponding to a higher/lower pitch, it results in bluer/redder light. You can then use this information to gauge the motion of celestial bodies.
A: The central postulate of relativity is that the speed of light is constant in every frame of reference. In Newtonian physics, this is impossible; if light is traveling at $c$ in one frame of reference, then someone traveling at velocity $v$ in the opposite direction would see light traveling at $c+v$. This led Einstein to conclude that Newtonian physics had foundational assumptions that were incorrect, namely that temporal and spatial distances are constant in every frame of reference. In relativity, if Alice and Bob are moving relative to each other, Bob will measure that Alice's clocks as going more slowly, and her distances to be shortened.
Clearly, even in Newtonian physics, the two people will disagree as to what happens constitutes the "same place"; since Bob views Alice as moving, what she considers the same place, Bob considers to be different places. For instance, suppose Alice is on a train. If she gets up from her seat, walks around, then sits back down in her seat, then from her perspective, she ended up in the "same" place. But to Bob, the whole train moved while Alice was walking around, so she did not end up in the same place.  Relativity says that not only will they disagree about what is the "same place", but also what is the "same time".
So suppose Alice is on Earth, traveling at rest with respect to Earth, and Bob is out in space at rest with respect to the Cosmic Microwave Background. Alice observes three events: E1 is light be emitted in all direction, E2 is a photon being absorbed after traveling the same direction Earth is traveling, and E3 is a photon being absorbed after traveling in the direction opposite to Earth's direction. Alice observes E2 and E3 as being the same distance and same elapsed time relative to E2. Bob measures E3 to be less elapsed time, but he also measures E3 to be less distance, so he still measures the photon to be traveling at $c$.
A: The speed of a wave is constant regardless of the source velocity. Only the frequency of the wave is affected by the source velocity. See this answer for sound waves: Does sound waves pick up the speed of its source?
A: 
The speed of light is constant

is one of Einstein's postulates of relativity and says something profound that can take time to realise.
It means that the light is measured to have the same speed independent of the motion of the source.
So for example if a light was shone from a stationary spaceship we would receive it at $3\times 10^8$ m/s.  If the same spaceship is moving towards or away from us at half the speed of light, and shines the same light towards us - we would still receive it at $3\times 10^8$ m/s.
