The electronic structure of non-Kramers ${\mathrm{Tb}}^{3+}$ centers in single crystals of yttrium aluminum garnet (YAG) was studied using a new-generation high-frequency magnetic resonance spectrometer that allows measurements of electron paramagnetic resonance (EPR), electron spin echo (ESE), the photon echo on hyperfine components of Stark levels in zero magnetic field, and optically detected magnetic resonance (ODMR). It was created on the basis of a highly stable microwave bridge operating in the near-terahertz region (0.094 or 0.130 THz), both in continuous-wave (cw) and pulse modes, a nonresonant system for supplying microwave power to a sample, and a cryogen-free magneto-optical cryostat. EPR, ESE, and cw measurements at two frequencies allowed us to reliably and with high accuracy determine the parameters of ${\mathrm{Tb}}^{3+}$ centers that occupy yttrium dodecahedral positions in YAG: ${g}_{||}=15.8\ifmmode\pm\else\textpm\fi{}0.2$, ${g}_{\ensuremath{\perp}}\ensuremath{\approx}0$ (|| corresponds to one of the $\ensuremath{\langle}100\ensuremath{\rangle}$ crystal axes), the zero-field splitting $\mathrm{\ensuremath{\Delta}}=2.705\ifmmode\pm\else\textpm\fi{}0.005\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}1}$, and the hyperfine interaction constant $A=0.197\ifmmode\pm\else\textpm\fi{}0.005\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}1}$. In addition to the ${\mathrm{Tb}}^{3+}$ centers in a regular environment, EPR spectra with lower intensity and a resolved hyperfine structure were found for at least three types of additional ${\mathrm{Tb}}^{3+}$ centers. They have symmetry, $g$ factors, and hyperfine interaction constants which are close to those of the main ${\mathrm{Tb}}^{3+}$ centers but differ in the zero-field splitting parameters \ensuremath{\Delta} that strongly depend on the crystal field. They were ascribed to ${\mathrm{Tb}}^{3+}$ centers with nearby antisite ${\mathrm{Y}}_{\mathrm{Al}}$ defects. Similarity with several types of ${\mathrm{Ce}}^{3+}$ centers in YAG crystals, the types, and relative concentrations of antisite defects are discussed. For one of the additional ${\mathrm{Tb}}^{3+}$ centers, splitting of the energy levels in the zero field turned out to be close to an energy of 94 GHz microwave quanta, and intense echo signals were observed in weak magnetic fields and in the zero field corresponding to the EPR transitions between the hyperfine components of the ${\mathrm{Tb}}^{3+}$ spin levels. ODMR spectra of ${\mathrm{Tb}}^{3+}$ centers in YAG crystals, containing Ce and Tb, were obtained by monitoring the intensity of the ${\mathrm{Ce}}^{3+}$ luminescence, which implies the transfer of energy from ${\mathrm{Tb}}^{3+}$ ions to ${\mathrm{Ce}}^{3+}$ ions with conservation of spin. This result seems to be important, since these systems are of interest for quantum communication and computations. Taking into account that cerium does not have isotopes with a nuclear spin and that terbium has 100% $^{159}\mathrm{Tb}$ having nuclear spin $I=3/2$ and a sufficiently large nuclear magnetic moment, the ${\mathrm{Tb}}^{3+}\ensuremath{-}{\mathrm{Ce}}^{3+}$ system in garnet crystals can be promising for coherent information processing. ${\mathrm{Tb}}^{3+}$ ions can play a role of qubits and ${\mathrm{Ce}}^{3+}$ ion with a lifetime of about 50 ns, and almost unity quantum efficiency of optical transitions can be used as a single-ion readout arrangement. The prospects for using this system as hardware for quantum communication and computations are discussed.