What is the methodology to discover thousands of compounds from a complex mixture?
Most compounds can only be analytically quantified individually, so the first step in analyzing a complex mixture is typically to separated the different compounds from each other. For something like coffee, this would typically be done via different types of chromatography (i.e. gas (GC), liquid (HPLC), ion (IC)).  

As the separated compounds exit the chromatography column, they can then be analyzed. This is usually done in real time using a number of techniques; the most relevant in this case is mass spectrometry (MS) following GC or HPLC.  

In short, the primary techniques used to analyzed the compounds found in an organic sample like coffee would be GC-MS and HPLC-MS.