DESTINY⁺ is an upcoming JAXA Epsilon medium-class mission to fly by the Geminids meteor shower parent body (3200) Phaethon. It will be the world's first spacecraft to escape from a near-geostationary transfer orbit into deep space using a low-thrust propulsion system. In doing so, DESTINY⁺ will demonstrate a number of technologies that include a highly efficient ion engine system, lightweight solar array panels, and advanced asteroid flyby observation instruments. These demonstrations will pave the way for JAXA's envisioned low-cost, high-frequency space exploration plans. Following the Phaethon flyby observation, DESTINY⁺ will visit additional asteroids as its extended mission. The mission design is divided into three phases: a spiral-shaped apogee-raising phase, a multi-lunar-flyby phase to escape Earth, and an interplanetary and asteroids flyby phase. The main challenges include the optimization of the many-revolution low-thrust spiral phase under operational constraints; the design of a multi-lunar-flyby sequence in a multi-body environment; and the design of multiple asteroid flybys connected via Earth gravity assists. This paper shows a novel, practical approach to tackle these complex problems, and presents feasible solutions found within the mass budget and mission constraints. Among them, the baseline solution is shown and discussed in depth; DESTINY⁺ will spend two years raising its apogee with ion engines, followed by four lunar gravity assists, and a flyby of asteroids (3200) Phaethon and (155140) 2005 UD. Finally, the flight operations plan for the spiral phase and the asteroid flyby phase are presented in detail.