Do sound waves travel through compression waves or longitudinal waves?
Sound waves are both compressional and longitudinal, (although we frequently draw them as transverse waves).
Longitudinal waves are where the particles of matter collide into one another, pushing them into the next and then oscillating back. This is how sound travels. This is also why sound can only travel through a medium, it cannot travel through a vacuum, as is shown here:
However, it is usually easier to draw the sound wave as a transverse wave - the compressions of the longitudinal wave become the antinodes of the sine wave, and the rarefactions become the nodes. An oscilloscope works in this manner.