Earthquake waves have several propagation speeds, because there are different types of waves. The fastest is called the P-wave, which is a compressional (longitudinal) wave, similar to a sound wave, with a velocity of ~5-8 km/s for typical continental bedrock. The second fastest is the S-wave, or shear wave, which is about 65% of the P-wave speed. These waves produce relatively little displacement at the surface (except for close to the epicenter of large earthquakes) but are important seismologically. Then, there are the surface waves, which are caused by the interaction of the S-waves with the surface (in a way that I don't 100% understand). These travel about 90% of the S-wave speed, but they have the biggest displacements at the surface and therefore are the main ones that you feel and that cause damage.

The surface wave displacements also get amplified in wet or loose soil, so the ground shaking and seismic damage is also much greater areas on top of sediment rather than bedrock. Areas on a river, lake or coast where the land has been extended into the water by dumping fill dirt are the worst--ground shaking is really bad and they are very prone to liquefaction.

The difference between the arrival times (at any given point on earth) of the different phases of seismic waves is a function of the distance from the earthquake itself (the hypocenter) and the observation site. It is close to linear in Euclidian distance relatively near the earthquake hypocenter, but becomes more nonlinear farther from the earthquake, because the wave speeds are faster at depth (denser rock) so the travel paths of the wave fronts (the ray paths) are nonlinear. These differences in arrival times are one of the main ways of locating the hypocenter of an earthquake given observations from seismometers at multiple sites. It's essentially triangulation, except with time differences instead of angles--this is done through solving a system of equations.

Additionally, S-waves can't pass through liquids, so there is the 'S-wave shadow zone' that occupies a large fraction of the side of the earth opposite an earthquake where there are no primary S-wave arrivals--S-waves are blocked by the liquid outer core. This is how we found out that the outer core is liquid!