The complex magnetism and transport properties of tetragonal Ce3Cu4As4O2 were examined through neutron scattering and physical property measurements on polycrystalline samples. The lamellar structure consists of alternating layers of CeCu4As4 with a single square Ce lattice and Ce2O2 bilayers. Peaks in the specific heat at the Néel temperature TN=24 K, T2=16 K, and T3=1.9 K indicate three magnetic phase transitions or distinct crossover phenomena. For T<TNneutron diffraction indicates the development of ferromagnetic ab sheets for both Ce sites, with alternating polarization along c, and a wave vector k1=c∗. For T<T2, quasi-two-dimensional low-energy spin fluctuations with k2=12a∗ and polarized perpendicular to k2 are suppressed. The data are consistent with quasi-two-dimensional antiferromagnetic order in the CeCu4As4 planes polarized along the k2 wave vector. T3 marks a spin-flop transition where the k1 staggered magnetization switches to in-plane polarization. There are significant transport anomalies associated with the transitions, in particular a substantial reduction in resistivity for T<TN. At T=100 mK the magnetic correlation length exceeds 75Å and the k1 modulated staggered moment is 0.85μB, which matches the 0.8μB saturation magnetization achieved for μ0H=7 T at T=2 K. We trace the unusual sequence of magnetic transitions to competing interactions and anisotropies in the alternating quasi-two-dimensional magnetic layers.