Exoplanets can evolve significantly between birth and maturity as their atmospheres, orbits, and structures are shaped by their environment. Young planets ($<$1Gyr)offertheopportunitytoprobethesesculptingprocesses.However,mostoftheknownyoungplanetsorbitprohibitivelyfaintstars.WepresentthediscoveryoftwoplanetstransitingHD63433(TOI1726,TIC130181866),ayoungSun-like($M_*=0.99\pm0.03$)star.Throughkinematics,lithiumabundance,androtation,weconfirmthatHD63433isamemberoftheUrsaMajormovinggroup($\tau=414\pm23$ Myr). Based on the TESS light curve and updated stellar parameters, the planet radii are $2.15\pm 0.10R\oplus$ and $2.67\pm 0.12R\oplus$, the orbital periods are 7.11 and 20.55 days, and the orbital eccentricities are lower than abut 0.2. Using HARPS-N velocities, we measure the Rossiter-McLaughlin signal of the inner planet, demonstrating the orbit is prograde. Since the host star is bright (V=6.9), both planets are amenable to transmission spectroscopy, radial velocity measurements of their masses, and more precise determination of the stellar obliquity. This system is therefore poised to play an important role in our understanding of planetary system evolution in the first billion years after formation.