Magnetization and ultrasound measurements have been performed on a TbFe₅Al₇ single crystal (tetragonal crystal structure) in the temperature range from 2 to 260 K in steady magnetic fields up to 18 T and in pulsed magnetic fields up to 60 T. The compound is a ferrimagnet (T _C=242 K) having an easy-plane anisotropy. Strong anisotropy is also present within the basal plane. At 2 K, the easy magnetization direction is the [1 0 0] axis. In the vicinity of the compensation temperature, T _comp=84 K, TbFe₅Al₇ displays a spin-reorientation transition from [1 0 0] to the [1 1 0] axis accompanied by pronounced anomalies in the relative sound-velocity change and sound attenuation. Further, field-induced magnetic transitions have been observed in TbFe₅Al₇ by magnetization and acoustic measurements. Step-wise rotation of the magnetic moments with a wide hysteresis occurs for fields applied along the [1 0 0] axis at T<T_comp and along the [1 1 0] axis at T>T_comp. The relative sound-velocity change displays sharp minima and the sound attenuation sharp maxima at the transitions. The critical field of the transitions tends to zero near the compensation point and grows sharply away from it reaching 19 and 33 T for fields applied along the [1 0 0] and [1 1 0] directions, respectively. The Tb-Fe inter-sublattice exchange constant has been determined directly from the high-field data and using molecular-field theory.