The masses, atmospheric makeups, spin-orbit alignments, and system architectures of extrasolar planets can be best studied when the planets orbit bright stars. We report the discovery of three bodies orbiting HD 106315, a bright (V = 8.97 mag) F5 dwarf targeted by our K2 survey for transiting exoplanets. Two small transiting planets are found to have radii and and orbital periods 9.55 days and 21.06 days, respectively. A radial velocity (RV) trend of 0.3 ± 0.1 m s-1 day-1 indicates the likely presence of a third body orbiting HD 106315 with period 160 days and mass 45 M ⊕. Transits of this object would have depths 0.1% and are definitively ruled out. Although the star has v sin i = 13.2 km s-1, it exhibits a short-timescale RV variability of just 6.4 m s-1. Thus, it is a good target for RV measurements of the mass and density of the inner two planets and the outer object's orbit and mass. Furthermore, the combination of RV noise and moderate v sin i makes HD 106315 a valuable laboratory for studying the spin-orbit alignment of small planets through the Rossiter-McLaughlin effect. Space-based atmospheric characterization of the two transiting planets via transit and eclipse spectroscopy should also be feasible. This discovery demonstrates again the power of K2 to find compelling exoplanets worthy of future study.