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Pd$_3$Bi$_2$Se$_2$ is a rare realization of a superconducting metal with a non-zero $Z_2$ topological invariant. We report the growth of high-quality single crystals of layered Pd$_3$Bi$_2$Se$_2$ with a superconducting transition at $T_c$ ~ 0.80 K and upper critical fields of ~10 mT and ~5 mT for the in-plane and out-of-plane directions, respectively. Our density functional theory (DFT) calculations reveal three pairs of doubly degenerate bands crossing the Fermi level, all displaying clear three-dimensional dispersion consistent with the overall low electronic anisotropy (<2). The multiband electronic nature of Pd$_3$Bi$_2$Se$_2$ is evident in magneto-transport measurements, yielding a sign-changing Hall resistivity at low temperatures. The magnetoresistance is non-saturating and follows Kohler's scaling rule. We interpret the magneto-transport data in terms of open orbits that are revealed in the DFT-calculated Fermi surface. de Haas-van Alphen (dHvA) oscillation measurements using torque magnetometry on single crystals yield four frequencies for out-of-plane fields: $F_\alpha = (150 \pm 26)$T, $F_\beta = (293 \pm 10)$T, $F_\gamma = (375 \pm 20)$T, and $F_\eta = (1017 \pm 12)$T, with the low frequency dominating the spectrum. Through the measurement of angular dependent dHvA oscillations and DFT calculations, we identify the $F_\alpha$ frequency with an approximately ellipsoidal electron pocket centered on the $L_2$ point of the Brillouin zone. Lifshitz-Kosevich analysis of the dHvA oscillations reveals a small cyclotron effective mass: $m^* = (0.11 \pm 0.02) m_0$ and a nontrivial Berry phase for the dominant orbit. The presence of nontrivial topology in a bulk superconductor positions Pd$_3$Bi$_2$Se$_2$ as a potential candidate for exploring topological superconductivity., Comment: 30 pages, 5 figures