Faster waves will travel the distance quicker and show up on the seismogram first. Travel time is a relative time, it is the number of minutes, seconds, etc. that the wave took to complete its journey. The precise speed that a seismic wave travels depends on several factors, most important is the composition of the rock. We are fortunate that the speed depends on the rock type because it allows us to use observations recorded on seismograms to infer the composition or range of compositions of the planet. But the process isn't always simple, because sometimes different rock types have the same seismic-wave velocity, and other factors also affect the speed, particularly temperature and pressure.
Temperature tends to lower the speed of seismic waves and pressure tends to increase the speed.
Pressure increases with depth in Earth because the weight of the rocks above gets larger with increasing depth. Usually, the effect of pressure is the larger and in regions of uniform composition, the velocity generally increases with depth, despite the fact that the increase of temperature with depth works to lower the wave velocity. When I describe the different seismic wave types below I'll quote ranges of speed to indicate the range of values we observe in common terrestrial rocks. But you should keep in mind that the specific speed throughout Earth will depend on composition, temperature, and pressure. The velocity of a wave depends on the elastic properties and density of a material.