Ultracold neutral plasmas provide a platform for studying transport properties in an idealized environment. In this dissertation, transport properties in a Ca$^+$/Yb$^+$ dual species ultracold neutral plasma and a Ca$^+$ magnetized ultracold neutral plasma are studied. In dual species plasmas, we study ion-ion temperature relaxation. We compare measured relaxation rates with atomistic simulations and a range of popular theories. Our work validates the assumptions and capabilities of molecular dynamic simulations and invalidates theoretical models in this regime. This work illustrates an approach for precision determinations of detailed material properties in Coulomb mixtures across a wide range of conditions. We also study plasma expansion in single species plasma in the presence of a strong uniform magnetic field. We find that the asymptotic expansion velocity falls exponentially with magnetic field strength, which disagrees with a previously published ambipolar diffusion model. In the parallel direction, plasma expansion is driven by electron pressure. However, in the perpendicular direction, no plasma expansion is observed at large magnetic field strengths.