What is the process involved in performing a mass spectrometry experiment?
This is hard to answer, because in my experience mass spectrometry isn't that time consuming at all. The experiment itself doesn't take that long. In this experiment the sample is ionized and charged fragments are accelerated in an electrical or magnetic field. Depending on their mass/charge ratio they will end up on specific locations in the detector.

The process around the experiment itself is dependent on your specific lab, but typically the mass spectrometer is operated by only a few people that do all the samples of the entire department. Because of that, there will often be a queue which could be several days. Additionally, not all mass spectra can be digitally identified as coming from a certain chemical, especially for larger molecules there are often various options that require human interference for the identification. In our lab this human interpretation is left to the person owning the sample, but if that part is also done by the mass spectrometer operators (I know in some labs this is the case) then I can imagine that this last step will be time-limiting.

To summarize: the experiment is short, the procedure around it (dependent on your specific lab) can take quite some time.