Tracing nuclear inflows and outflows in AGNs, determining the mass of gas involved in them, and their impact on the host galaxy and nuclear black hole requires 3-D imaging studies of both the ionized and molecular gas. We map the distribution and kinematics of molecular and ionized gas in a sample of active galaxies to quantify the nuclear inflows and outflows. Here, we analyze the nuclear kinematics of NGC 1566 via ALMA observations of the CO J:2-1 emission at 24pc spatial and ~2.6km/s spectral resolution, and Gemini-GMOS/IFU observations of ionized gas emission lines and stellar absorption lines at similar spatial resolution, and 123km/s of intrinsic spectral resolution. The morphology and kinematics of stellar, molecular (CO), and ionized ([NII]) emission lines are compared to the expectations from rotation, outflows, and streaming inflows. While both ionized and molecular gas show rotation signatures, there are significant non-circular motions in the innermost 200pc and along spiral arms in the central kpc (CO). The nucleus shows a double-peaked CO profile (full width at zero intensity of 200km/s), and prominent (~80km/s) blue- and redshifted lobes are found along the minor axis in the inner arcseconds. Perturbations by the large-scale bar can qualitatively explain all features in the observed velocity field. We thus favor the presence of a molecular outflow in the disk with true velocities of ~180km/s in the nucleus and decelerating to 0 by ~72pc. The implied molecular outflow rate is 5.6M_{Sun}_/yr, with this gas accumulating in the nuclear 2" arms. The ionized gas kinematics support an interpretation of a similar but more spherical outflow in the inner 100pc, with no signs of deceleration. There is some evidence of streaming inflows of ~50km/s along specific spiral arms, and the estimated molecular mass inflow rate, ~0.1M_{Sun}_/yr, is significantly higher than the SMBH accretion rate (dM/dt=4.8x10^-5^M_{Sun}_/yr).