This paper studied the structural and luminescent characteristics of undoped and doped-with-Tb3+-ions ZnO films of 200 nm and 600 nm thicknesses, grown via RF magnetron sputtering on (100) silicon substrate in Ar and Ar-O2 plasma. X-ray diffraction (XRD) patterns revealed a strong preferred orientation of ZnO and ZnO:Tb crystals of the wurtzite structure along the c-axis, perpendicular to the substrate. In the as-deposited ZnO:Tb films, the additional crystal phases, namely, Tb2O3, TbO2, and an amorphous phase, were revealed. The as-deposited undoped films were under tensile strain, that increased in the doped films. This proved the incorporation of the Tb3+ ions into the ZnO grains, and agreed with the Raman spectra investigation. The XRD data and atomic force microscopy study showed that Tb doping impeded the growth of grains and columns, respectively. The photoluminescence (PL) spectra of the doped films contained the UV band ascribed to exciton PL, a broad intrinsic defect-related band, and the narrow bands caused by the intra-shell transitions of the Tb3+ ions. Terbium doping suppressed ZnO emissions. The post-deposition rapid thermal annealing at up to 800 °C of both the undoped and doped films promoted tensile strain relaxation, grain growth, improvement in the ZnO crystal structure, and an increase in the exciton PL. The intensity of the Tb3+ PL changed non-monotonically, and was the highest for the film annealed at 600 °C. The conventional thermal annealing promoted the non-monotonic changes in the strains and grain sizes in such a way that, after annealing at 900 °C, their values became the same as in the as-deposited ZnO:Tb film. This structural change was accompanied by a decrease in the exciton and Tb3+ PL intensity. The formation of the Zn2SiO4 phase was observed via XRD, and confirmed via scanning electron microscopy. It was attributed to the interdiffusion through the film/substrate interface. The deposition in the Ar-O2 atmosphere is found to be more preferable for the formation of Tb3+ emission centers in the ZnO matrix.