Over 4000 exoplanets have been identified and thousands of candidates are to be confirmed. The relations between the characteristics of these planetary systems and the kinematics, Galactic components, and ages of their host stars have yet to be well explored. To address these questions, we conduct a research project, dubbed Planets Across Space and Time (PAST). To do this, one of the key steps is to accurately characterize the planet host stars. In this paper, Paper I of the PAST series, we revisit the kinematic method for classification of Galactic components and extend the applicable range of velocity ellipsoid from ~100pc to ~1500pc from the Sun in order to cover most known planet hosts. Furthermore, we revisit the age-velocity dispersion relation (AVR), which allows us to derive kinematic ages with a typical uncertainty of 10-20% for an ensemble of stars. Applying the above revised methods, we present a catalog of kinematic properties (i.e., Galactic positions, velocities, and the relative membership probabilities among the thin disk, thick disk, Hercules stream, and the halo) as well as other basic stellar parameters for 2174 host stars of 2872 planets by combining data from Gaia, LAMOST, APOGEE, RAVE, and the NASA exoplanet archive. The revised kinematic method and AVR, as well as the stellar catalog of kinematic properties and ages, lay the foundation for future studies of exoplanets in space and time in the Galactic context.