We present results from the first three years of observations of moving group targets in the SEEDS high-contrast imaging survey of exoplanets and disks using the Subaru telescope. We achieve typical contrasts of ~10^5 at 1 and ~10^6 beyond 2 around 63 proposed members of nearby kinematic moving groups. We review each of the kinematic associations to which our targets belong, concluding that five, beta Pictoris (~20 Myr), AB Doradus (~100 Myr), Columba (~30 Myr), Tucana-Horogium (~30 Myr), TW Hydrae (~10 Myr), are sufficiently well-defined to constrain the ages of individual targets. Somewhat less than half of our targets are high-probability members of one of these moving groups. For all of our targets, we combine proposed moving group membership with other age indicators where available, including Ca II HK emission, X-ray activity, and rotation period, to produce a posterior probability distribution of age. SEEDS observations discovered a substellar companion to one of our targets, kappa And, a late B star. We do not detect any other substellar companions, but do find seven new close binary systems, of which one still needs to be confirmed. A detailed analysis of the statistics of this sample, and of the companion mass constraints given our age probability distributions and exoplanet cooling models, will be presented in a forthcoming paper.