We have used the Gemini Near-Infrared Integral Field Spectrograph in the J and K bands to map the distribution, excitation, and kinematics of the ionized H ii and warm molecular gas H2, in the inner few 100 pc of six nearby active galaxies: NGC 788, Mrk 607, NGC 3227, NGC 3516, NGC 5506, NGC 5899. For most galaxies, this is the first time that such maps have been obtained. The ionized and H2 gas show distinct kinematics: while the H2 gas is mostly rotating in the galaxy plane with low velocity dispersion (σ), the ionized gas usually shows signatures of outflows associated with higher σ values, most clearly seen in the [Fe ii] emission line. These two gas species also present distinct flux distributions: the H2 is more uniformly spread over the whole galaxy plane, while the ionized gas is more concentrated around the nucleus and/or collimated along the ionization axis of its active galactic nucleus (AGN), presenting a steeper gradient in the average surface mass density profile than the H2 gas. The total H ii masses cover the range 2×105--2×107 M⊙, with surface mass densities in the range 3–150 M⊙ pc−2, while for the warm H2 the values are 103–4 times lower. We estimate that the available gas reservoir is at least ≈ 100 times more massive than needed to power the AGN. If this gas forms new stars the star formation rates, obtained from the Kennicutt–Schmidt scaling relation, are in the range 1–260 × 10−3 M⊙ yr−1. But the gas will also – at least in part – be ejected as the observed outflow