I am not really a "budding seismologist." I am a chemistry/physics teacher teaching an Earth science class (low academic level) in a small-town high school. (My college coursework in earth sciences = 0) I understand the concept of triangulation to determine an earthquake's origin, and I realize there are some subtleties in wave analysis such as pP and sP waves. What I don't understand is how the triangulation determines the epicenter instead of the focus. Wouldn't the S-P lag times for 3 stations give an intersection point at the focus since that is the actual point of origin for the waves?
This is a good question, but it really has a simple answer. How can you determine the depth of an object when you are not even calculating it? Triangulation uses circles, 2-D shapes, so when finding the solution to the equations of circles that solve all 3 circles, you only find X and Y. Z isn't even being solved for.
Triangulation is not precise. In practice, we end up using much more than 3 stations and statistical methods to find the precise epicenter (and other methods to find the focus).
The method of drawing circles on a map to find their intersection solves for only one unknown. When the stations are far from the earthquake focus, or the earthquake is shallow, the difference between the distance to the focus and epicenter is ignored. Because many earthquakes are shallow, the three circles often do intersect in a point. When they do not, a possible explanation is that it is because the earthquake is relatively deep (compared to the distance to the epicenter.)
If you want to try to determine both the epicenter and the depth of the earthquake, then you need to collect more observations (more stations) and use mathematics to solve a system having at least two unknowns. You might use linear algebra. Note that arrival times for stations closer to the earthquake will be affected by the depth of the focus than stations far, so you would want to try to include stations closer to the earthquake. Secondary school physics student might have had enough math to use the Newton-Raphson method method to estimate the depth. They might try to used the initial estimate of the 'epicenter' by the first method, to constrain the equations they try to solve to estimate a depth.
In practice, seismologists collect large amounts of data and have to consider many unknowns - such as distance to focus, but also the variation of seismic velocity through different types of rocks.
If you try the three-station and map approach in your Earth Science classroom, the students may notice that all three circles do not go though a single point typically, but come more or less close to doing so. Perhaps this is a good opportunity to discuss the differences between a model and actual earthquakes and Earth. It might even be possible to discuss how a better model can improve the agreement with observations.
It looks like this USGS page : Determining the Depth of an Earthquake has a somewhat different answer, which may describe how deep earthquakes were 'discovered' historically :)
This presentation at the Berkeley Seismographic Station shows a geometric construction and formula for determining earthquake depth.