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 TESS space mission in Sectors 5 and 6. We performed ground-based follow-up observations -- comprised of LCOGT transit photometry, NIRC2 adaptive optics imaging, and FIES, CORALIE, HARPS, HIRES, and PFS 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 5 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-421b, has an orbital period of Pb =5.19672 +- 0.00049 days, a mass of Mb = 7.17 +- 0.66 Mearth and a radius of Rb = 2.68+0.19-0.18 Rearth, whereas the outer warm Neptune, TOI-421 c, has a period of Pc =16.06819 +- 0.00035 days, a mass of Mc = 16.42+1.06-1.04 Mearth, a radius of Rc = 5.09+0.16-0.15 Rearth and a density of rho_c =0.685+0.080-0.072 g cm-3 . With its characteristics the inner planet (rho_b=2.05+0.52-0.41 g cm-3) is placed in the intriguing class of the super-puffy mini-Neptunes. TOI-421b and TOI-421c are found to be well suitable for atmospheric characterization. Our atmospheric simulations predict significant Ly-alpha transit absorption, due to strong hydrogen escape in both planets, and the presence of detectable CH_4 in the atmosphere of TOI-421c if equilibrium chemistry is assumed.
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