  This paper presents an in-depth exploration of timelike free geodesics in
spatially curved Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. A
unified approach for these geodesics encompassing both radial and non-radial
trajectories across Euclidean, spherical, and hyperbolic geometries is
employed. Using the symmetry properties of the system, two constants of motion
related to this dynamical system are derived. This treatment facilitates the
explicit computation of radial and angular peculiar velocities, along with the
evolution of the comoving radial distance and angle over time. The study
introduces three distinct methods for characterizing geodesic solutions,
further delving into the flatness limit, the null geodesic limit, the radial
geodesic limit, the comoving geodesic limit, and the circular orbits.
Additionally, we analyze Killing vectors, reflecting the symmetries inherent in
the dynamical system. This study provides a more profound understanding of the
behavior of free particles as observed from a comoving reference frame.
