We investigate the effect of rotation on the star formation process quantitatively using axisymmetric numerical calculations. An adiabatic hydrostatic object (the so-called first core) forms in a contracting cloud core, after the central region becomes optically thick and continues to contract, driven by mass accretion onto it. The structure of a rotating first core is characterized by its total angular momentum J_core and mass M_core, both of which increase by accretion with time. We find that the first core evolves with a constant J_core/M. Evolutionary paths of first cores can be classified into two types. In a slowly rotating core with J_core/M < 0.015G/(c_iso), where c_iso and G represent the isothermal sound speed in the molecular cloud core and the gravitational constant, respectively, the core begins "second collapse" after the central density exceeds the H₂ dissociation density. This is the same evolution as a standard scenario for a spherically symmetric, nonrotating core. On the other hand, a core with J_core/M > 0.015G/(c_iso) stops its contraction before the central density reaches the H₂ dissociation density and does not begin the second collapse. These rapidly rotating first cores suffer from nonaxisymmetric instabilities, such as formation of massive spiral arms, deformation into a bar, or fragmentation. Although the rotating first cores have small average luminosities of L_core = 0.003-0.03 L, assuming a constant mass accretion rate _core. Their lifetimes last several thousand years or more, which is much longer than those expected for nonrotating clouds (~1000 yr). We expect that at least several percent of prestellar cores contain first cores as very low luminosity objects. Furthermore, we find a core with 0.012G/(c_iso) < J_core/M < 0.015G/(c_iso) may form close binary systems with initial separation of 0.02-0.1 AU after the second collapse phase.