${\mathit{L}}_{3}$-edge x-ray absorption, inelastic neutron scattering (INS), and magnetic susceptibility experiments are used to characterize the electronic properties of Tb in ${\mathrm{Pb}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{TbCu}}_{3}$${\mathrm{O}}_{8}$. The x-ray absorption results indicate that Tb is trivalent and show no evidence for strong hybridization between Tb and the ${\mathrm{CuO}}_{2}$ bands. The observation of three peaks of magnetic origin in the INS experiments is also a clear indication of the trivalent state of Tb. These peaks are assigned to crystalline electric-field (CEF) transitions, and their energies and intensities are consistent with the CEF potential expected for a trivalent rare earth in the ${\mathrm{Pb}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{RCu}}_{3}$${\mathrm{O}}_{8}$ lattice. The low-energy response in the INS spectra at low temperatures is described using the mean-field approximation. Single crystal and polycrystalline susceptibilities show deviations from calculations and differ from magnetically diluted samples. These deviations are observed for temperatures less than 120 K and we interpret this as the onset temperature of the magnetic Tb-Tb short-range correlations. This interpretation is supported by the temperature dependence of the magnetic INS intensities at low energies. The onset temperature, estimated at 120 K, is more than an order of magnitude higher than seen in any related system, as well as much higher than the two-dimensional long-range ordering temperature at T=5.5 K. This unusual behavior is understood in terms of the highly two-dimensional nature of the lattice combined with the strongly Ising-like magnetic properties. In addition to the Tb-Tb correlations, the susceptibility experiments on a magnetically dilute sample indicate a weak rare-earth Cu exchange interaction.