We study the spin alignment of galaxies and halos with respect to filaments and walls of the cosmic web, identified with DisPerSE, using the SIMBA simulation from z=0-2. Massive halos' spins are oriented perpendicularly to their closest filament's axis and walls, while low mass halos tend to have their spins parallel to filaments and in the plane of walls. A similar mass-dependent spin flip is found for galaxies, albeit with a weaker signal particularly at low mass and low-z, suggesting that galaxies' spins retain memory of their larger-scale environment. Low-z star-forming and rotation-dominated galaxies tend to have spins parallel to nearby filaments, while quiescent and dispersion-dominated galaxies show preferentially perpendicular orientation; the star formation trend can be fully explained by the stellar mass correlation, but the morphology trend cannot. There is a strong dependence on HI mass, such that high-HI galaxies tend to have parallel spins while low-HI galaxies are perpendicular, which persists even when matching samples in stellar mass, suggesting that HI content traces anisotropic infall more faithfully than the stellar component. Finally, at fixed stellar mass, the strength of spin alignments correlates with the filament's density, with parallel alignment for galaxies in high density environments. These findings are consistent with conditional tidal torque theory, and highlight a significant correlation between galactic spin and the larger scale tides that are important e.g. for interpreting weak lensing studies. SIMBA allows us to rule out numerical grid locking as the cause of previously-seen low mass alignment.