you are very welcome, especially because this is a crucial point in geoscience.
Interestingly, there are many different types of seismic waves, that can be classified according to the kind of motion they induce to the material they propagate in. Some of them, for example, can propagate only in fluids and not in rocks, or the other way around, but some can actually travel all kind of materials, fluid, rocky, sandy, so we can actually measure these waves literally on the other side of the Earth, with respect to where they started by the earthquake. From this kind of measurements we can understand which materials have been in between the source and the seismograph we are looking at.
I found this sketch on the internet (at the very end of the page), I think it may help understanding how waves propagate in the different parts of the Earth interior: http://www.cyberphysics.co.uk/topics/earth/geophysics/Seismic%20Waves%20Reading.htm
You are right. The P waves (the primary waves and fastest) travel through all the layers. However, the S waves (secondary and slowest) do not travel through liquid. That is one of the reasons to suggest that the outer core is liquid. Because S waves are not recorded coming out of that region. P waves that travel through that region are slow. In addition, modelling of temperature, pressure and minerals at that depth help to suggest what is down there.