I know the focus of an earthquake is where the earthquake originated from, but what I could never figure out is, how to scientists find out where exactly the focus (and epicenter) are located?
Earthquake epicenters are located using triangulation, this is possible once seismograms of the earthquake - coming from at least three locations - have been analyzed properly. Here is a good explanation on a site for seismology students at Michigan Tech which takes its seismogram illustrations from Bolt's textbook on earthquakes (1978). Read this page and you will have a good explanation of how seismologists determine the location of an earthquake epicenter.
The seismometer records the time when the P and S-waves arrive at the recording station. P-waves travel faster through the earth than S-waves and so they arrive at the seismometer station before the S-waves and are recorded by the seismometer first. The difference in arrival time between the two types of seismic wave can be used to calculate the distance of the earthquake's epicenter from the seismometer, as the further away an earthquake is, the greater the lag time between the detection of the S waves relative to the P waves.