If I were to stand at one end of a light-year long metal pole, and another person were to stand one light-year away at the other end, and then I were to push on my end of the pole. How long would it take for their side of the pole to move? I've been told that the matter in the pole would only move at the speed of sound. So it would take however long a sound-wave takes to travel a light-year for the other person to see the pole move. But if this is true, does that mean that if you were to take any still object, and hit it with another object moving at the speed of sound, the still object would compress into itself before expanding back out again?
You are quite correct, the movement of the pole would travel at the speed of sound in whatever the pole is made of, and this does apply to any object.
Although I suspect it's not the real point of your question, the speed of sound in steel is about 6100 m/s, so assuming your pole is made from steel it would take about 49,200 years for the other end of the pole to move.