The discovery of planets around other stars is revolutionizing our notions of planet formation and is poised to do the same for planetary climate. Studying transiting planets is complementary to eventual studies of directly imaged planets: (1) we can readily measure the mass and radius of transiting planets, linking atmospheric properties to bulk composition and formation, (2) many transiting planets are strongly irradiated and exhibit novel atmospheric physics, and (3) the most common temperate terrestrial planets orbit close to red dwarf stars and are difficult to image directly. We have only been able to comprehensively characterize the atmospheres of a handful of transiting planets, because most orbit faint stars. The Transiting Exoplanet Survey Satellite (TESS) will discover transiting planets orbiting the brightest stars, enabling, in principle, an atmospheric survey of 102–103 bright hot Jupiters and warm sub-Neptunes. Uniform observations of such a statistically significant sample would provide leverage to understand—and learn from—the diversity of short-period planets, and would identify the minority of truly special planets worthy of more intensive follow-up. We argue that the best way to maximize the scientific returns of TESS is to adopt a triage approach. A space mission consisting of a ∼1 m telescope with an optical–NIR spectrograph could measure molecular absorption for nonterrestrial planets discovered by TESS, as well as eclipses and phase variations for the hottest jovians. Such a mission could observe up to 103 transits per year, thus enabling it to survey a large fraction of the bright (J < 11) hot-Jupiters and warm sub-Neptunes TESS is expected to find. The James Webb Space Telescope (JWST) could be used to perform detailed atmospheric characterization of the most interesting transiting targets (transit, eclipse, and—when possible—phase-resolved spectroscopy). TESS is also expected to discover a few temperate terrestrial planets transiting nearby M-Dwarfs. Characterizing these worlds will be time-intensive: JWST will need months to provide tantalizing constraints on the presence of an atmosphere, planetary rotational state, clouds, and greenhouse gases. Future flagship missions should be designed to provide better constraints on the habitability of M-Dwarf temperate terrestrial planets.