The 2013 Planetary Science Decadal Survey identified a detailed investigation of the Trojan asteroids occupying Jupiters L4 and L5 Lagrange points as a priority for future NASA missions. Observing these asteroids and measuring their physical characteristics and composition would aid in identification of their source and provide answers about their likely impact history and evolution, thus yielding information about the makeup and dynamics of the early Solar System. We present a conceptual design for a mission to the Jovian Trojan asteroids: the Trojan ASteroid Tour, Exploration, and Rendezvous (TASTER) mission, that is consistent with the NASA New Frontiers candidate mission recommended by the Decadal Survey and the final result of the 2011 NASA-JPL Planetary Science Summer School. Our proposed mission includes visits to two Trojans in the L4 population: a 500 km altitude fly-by of 1999 XS143, followed by a rendezvous with and detailed observations of 911 Agamemnon at orbital altitudes of 1000 - 100 km over a 12 month nominal science data capture period. Our proposed instrument payload - wide- and narrow-angle cameras, a visual and infrared mapping spectrometer, and a neutron/gamma ray spectrometer - would provide unprecedented high-resolution, regional-to-global datasets for the target bodies, yielding fundamental information about the early history and evolution of the Solar System. Although our mission design was completed as part of an academic exercise, this study serves as a useful starting point for future Trojan mission design studies. In particular, we identify and discuss key issues that can make large differences in the complex trade-offs required when designing a mission to the Trojan asteroids.