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 Echellé Spectrograph, CORALIE, High Accuracy Radial velocity Planet Searcher, High Resolution Échelle 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⊕, and a radius of R_b = 2.68^(+0.19)_(-0.18) R⊕, whereas the outer warm Neptune, TOI-421 c, has a period of Pc = 16.06819 ± 0.00035 days, a mass of M_c = 16.42^(+1.06)_(-1.04) M⊕, a radius of R_c = 5.09^(+0.16)_(-0.15) R⊕ and a density of ρ_c = 0.685^(+0.080)_(-0.072) g cm⁻³. With its characteristics, the outer planet (ρ_c = 0.685^(+0.080)_(-0.072) g 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 Lyα 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
