We present self-consistent N-body/hydrodynamical simulations of isolated, Milky Way-like galaxies with stellar and gaseous disks. By varying the initial gas fraction, we find that gas delays bar formation and, in the most gas-rich model, suppresses it by exciting higher-order spiral modes and promoting early central mass growth. Once a bar forms, continued gas inflow and gravitational torques weaken the bar and inhibit its secular slowdown, keeping bars in the gas-bearing models shorter and predominantly in the fast-bar regime, while the gas-free bar evolves into the slow-bar regime. Stars and gas respond differently to the same barred potential. While the mean stellar velocity field is well described by a smooth bisymmetric pattern, the cold and warm gas follow elongated, x₁-like streamlines with sharp shock-induced velocity transitions, producing sawtooth-like radial and archlike tangential velocity profiles. Despite these strong noncircular motions, their azimuthally averaged tangential velocities closely follow the circular velocity over most of the disk. By contrast, the hot gas is dominated by stellar feedback and thermal pressure and is largely decoupled from the ordered bar-driven flow. To describe these bar-driven gas flows, we introduce the barflow model, a phenomenological description of non-sinusoidal gas streaming motions with a radially varying shock-front phase. Mock line-of-sight velocity-fitting experiments show that the barflow model recovers the intrinsic rotation curve, bar-driven velocity perturbations, and radial gas transport more accurately than the conventional bisymmetric model.