We report a detailed investigation of quantum oscillations in Sr₃Ru₂O₇, observed inductively (the de Haas--van Alphen effect) and thermally (the magnetocaloric effect). Working at fields from 3 to 18 T allowed us to straddle the metamagnetic transition region and probe the low- and high-field Fermi liquids. The observed frequencies are strongly field dependent in the vicinity of the metamagnetic transition, and there is evidence for magnetic breakdown. We also present the results of a comprehensive rotation study. The most surprising result concerns the field dependence of the measured quasiparticle masses. Contrary to conclusions previously drawn by some of us as a result of a study performed with a much poorer signal-to-noise ratio, none of the five Fermi-surface branches for which we have good field-dependent data gives evidence for a strong-field dependence of the mass. The implications of these experimental findings are discussed.Instituto de Física La PlataInstituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas