Polycrystalline samples of RE2O2CO3 (RE = Pr, Nd, Gd, Tb, Dy, Ho, Er, and Yb) with a unique rare-earth bilayer of triangular lattice were synthesized and studied by DC and AC magnetic susceptibility. Data reveals various magnetic ground states including (i) a nonmagnetic ground state for the Pr sample; (ii) long range magnetic ordering for the Nd, Gd, Tb, Dy, Ho, and Er samples. Besides the Gd sample, they exhibit field-induced spin state transitions. More interestingly, the series spin state transitions in the Nd and Dy samples could be attributed to the field-induced up-up-down (UUD) spin structure. Neutron powder diffraction (NPD) measurements of the Er sample suggest a spiral spin structure below its TN; and (iii) a short-range ordering for the Yb sample. The disrupted inter-layer interaction due to the shift of Yb3+ ions within the bilayer prevents long range magnetic ordering down to 30 mK and makes it another Yb-related triangular lattice antiferromagnet that has the potential to realize a quantum spin liquid state.