We report the discovery of a warm Neptune and a hot sub-Neptune transiting TOI-421 (BD-14 1137, TIC 94986319), a bright (V = 9.9) G9 dwarf star in a visual binary system observed by the Transiting Exoplanet Survey Satellite (TESS) space mission in Sectors 5 and 6. We performed ground-based follow-up observations-comprised of Las Cumbres Observatory Global Telescope transit photometry, NIRC2 adaptive optics imaging, and FIbre-fed Echelle Spectrograph, CORALIE, High Accuracy Radial velocity Planet Searcher, High Resolution echelle Spectrometer, and Planet Finder Spectrograph high-precision Doppler measurements-and confirmed the planetary nature of the 16 day transiting candidate announced by the TESS team. We discovered an additional radial velocity signal with a period of five days induced by the presence of a second planet in the system, which we also found to transit its host star. We found that the inner mini-Neptune, TOI-421 b, has an orbital period of P-b = 5.19672 +/- 0.00049 days, a mass of M-b = 7.17 +/- 0.66 M-circle plus, and a radius of R-b = R-circle plus, whereas the outer warm Neptune, TOI-421 c, has a period of P-c = 16.06819 +/- 0.00035 days, a mass of M-c = 16.42(-1.04)(+1.06)M(circle plus), a radius of R-c = 5.09(-0.15)(+0.16)R(circle plus), and a density of rho(c) = 0.685(-0.072)(+0.080) cm(-3). With its characteristics, the outer planet (rho(c) = 0.685(-0.0072)(+0.080) cm(-3)) is placed in the intriguing class of the super-puffy mini-Neptunes. TOI-421 b and TOI-421 c are found to be well-suited for atmospheric characterization. Our atmospheric simulations predict significant Ly alpha transit absorption, due to strong hydrogen escape in both planets, as well as the presence of detectable CH4 in the atmosphere of TOI-421 c if equilibrium chemistry is assumed.