One bottleneck for the exploitation of data from the Kepler mission for stellar astrophysics and exoplanet research has been the lack of precise radii and evolutionary states for most of the observed stars. We report revised radii of 177911 Kepler stars derived by combining parallaxes from the Gaia Data Release 2 with the DR25 Kepler Stellar Properties Catalog. The median radius precision is ~8%, a typical improvement by a factor of 4-5 over previous estimates for typical Kepler stars. We find that ~67% (~120000) of all Kepler targets are main-sequence stars, ~21% (~37000) are subgiants, and ~12% (~21000) are red giants, demonstrating that subgiant contamination is less severe than some previous estimates and that Kepler targets are mostly main-sequence stars. Using the revised stellar radii, we recalculate the radii for 2123 confirmed and 1922 candidate exoplanets. We confirm the presence of a gap in the radius distribution of small, close-in planets, but find that the gap is mostly limited to incident fluxes >200F_{Earth}_, and its location may be at a slightly larger radius (closer to ~2R_{Earth}_) when compared to previous results. Furthermore, we find several confirmed exoplanets occupying a previously described "hot super-Earth desert" at high irradiance, show the relation between a gas-giant planet's radius and its incident flux, and establish a bona fide sample of eight confirmed planets and 30 planet candidates with Rp<2R_{Earth}_ in circumstellar "habitable zones" (incident fluxes between 0.25 and 1.50F_{Earth}_). The results presented here demonstrate the potential for transformative characterization of stellar and exoplanet populations using Gaia data.