Characterizing Transiting Planet Atmospheres through 2025

[Abridged] We have only been able to comprehensively characterize the atmospheres of a handful of transiting planets, because most orbit faint stars. TESS will discover transiting planets orbiting the brightest stars, enabling, in principle, an atmospheric survey of 10^2 to 10^3 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. We argue that the best way to maximize the scientific returns of TESS is with a follow-up space mission consisting of a ~1 m telescope with an optical--NIR spectrograph: it could measure molecular absorption for non-terrestrial planets, as well as eclipses and phase variations for the hottest jovians. Such a mission could observe up to 10^3 transits per year, thus enabling it to survey a large fraction of the bright (J<11) TESS planets. 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.