We present a new observational project to study the hierarchical triple stellar system Algol, concentrating on the semidetached eclipsing binary at the heart of the system. Over 140 high-resolution and high-S/N spectra have been secured, of which 80 are from FIES at the Nordic Optical Telescope, La Palma, and the remainder were obtained with BOES at the Bohyunsan Optical Astronomy Observatory in Korea. All three components were successfully detected by the method of spectral disentangling, which yields the individual spectra of the three stars and also high-quality spectroscopic elements for both the inner and outer orbits. We present a detailed abundance study for the mass-accreting component in the inner orbit, which holds information on the history of mass transfer in the close inner binary system. We also reveal the atmospheric parameters and chemical composition of the tertiary component in the outer orbit.