We introduce the first two simulations of the IllustrisTNG project, a next generation of cosmological magnetohydrodynamical simulations, focusing on the optical colors of galaxies. We explore TNG100, a rerun of the original Illustris box, and TNG300, which includes 2x2500^3 resolution elements in a volume twenty times larger. Here we present first results on the galaxy color bimodality at low redshift. Accounting for the attenuation of stellar light by dust, we compare the simulated (g-r) colors of 10^9 < M*/Msun < 10^12.5 galaxies to the observed distribution from the Sloan Digital Sky Survey (SDSS). We find a striking improvement with respect to the original Illustris simulation, as well as excellent quantitative agreement in comparison to the observations, with a sharp transition in median color from blue to red at a characteristic M* ~ 10^10.5 Msun. Investigating the build-up of the color-mass plane and the formation of the red sequence, we demonstrate that the primary driver of galaxy color transition in the TNG model is supermassive blackhole feedback in its low-accretion state. Across the entire population we measure a median color transition timescale dt_green of ~1.6 Gyr, a value which drops for increasingly massive galaxies. We find signatures of the physical process of quenching: at fixed stellar mass, redder galaxies have lower SFRs, gas fractions, and gas metallicities; their stellar populations are also older and their large-scale interstellar magnetic fields weaker than in bluer galaxies. Finally, we measure the amount of stellar mass growth on the red sequence. Galaxies with M* > 10^11 Msun which redden at z<1 accumulate on average ~25% of their final z=0 mass post-reddening; at the same time, ~18% of such massive galaxies acquire half or more of their final stellar mass while on the red sequence.