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. Its 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.