In RbCoCl₃, the magnetic Co²⁺ ions form weakly coupled Ising chains arranged on a triangular lattice. We have investigated the structure and magnetism in RbCoCl₃ by high-resolution x-ray diffraction and neutron scattering measurements on powder and single-crystal samples between 1.5 and 300 K. Upon cooling, the Co²⁺ spins develop one-dimensional antiferromagnetic correlations along the chain axis (c axis) below 90 K. Below the first Néel temperature, T_N1 = 28 K, a partial three-dimensional magnetic order sets in, with propagation vector k₁ = (1/3, 1/3, 1), the moments aligned along the c axis, and every third chain uncorrelated from its neighbors. Only below a second magnetic phase transition at T_N2 = 13 K does the system achieve a fully ordered state, with two additional propagation vectors: k₂ = (0, 0, 1) establishes a "honeycomb" c-axis order, in which one third of the chains are subject to a strong effective mean field due to their neighbors whereas two thirds experience no net field, while k₃ = (1/2, 0, 1) governs a small, staggered, in-plane ordered moment. We conclude that RbCoCl₃ is an excellent material to study the physics of Ising chains in a wide variety of temperature-controlled environments, and our results build an important foundation for the accurate interpretation of the spin dynamics measured in each case.