ALMA's high spatial resolution and sensitivity have recently begun to reveal previously unresolved sub-structure in protoplanetary disks, simultaneously shedding light and posing new questions on disk evolution. AA Tau hosts a highly inclined protoplanetary disk, and is the prototypical source for a class of stars with a peculiar periodic photometric variability thought to be dependent on viewing geometry. We present high resolution (~0.2") ALMA observations of the 1.3mm dust continuum, which surprisingly reveal a multi-ringed transition disk at an inclination of ~60 degrees. In addition to the ringed sub-structure, we find non-axisymmetric features, including a 'bridge' of flux across the inner-most gap. Observations of HCO+ 2-1 show bright emission from within the inner-most gap, suggesting that large quantities of molecular gas still reside within the hole. The kinematics of this emission are indicative of a radial flow, possibly associating the gas with gap-crossing streamers. AA Tau may therefore offer a rare glimpse into the dynamical processes governing accretion in planet-forming transition disks. We additionally discuss how resolving the disk geometry and substructure may affect interpretations of the mechanism behind AA Tau's variability, and speculate on the nature of the continuum 'flux bridge'.