A simultaneously transmitting and reflecting surface (STARS) aided terahertz (THz) communication system is proposed. A novel power consumption model is proposed that depends on the type and resolution of the STARS elements. The spectral efficiency (SE) and energy efficiency (EE) are maximized in both narrowband and wideband THz systems by jointly optimizing the hybrid beamforming at the base station (BS) and the passive beamforming at the STARS. 1) For narrowband systems, independent phase-shift STARSs are investigated first. The resulting complex joint optimization problem is decoupled into a series of subproblems using penalty dual decomposition. Low-complexity element-wise algorithms are proposed to optimize the analog beamforming at the BS and the passive beamforming at the STARS. The proposed algorithm is then extended to the case of coupled phase-shift STARS. 2) For wideband systems, the spatial wideband effect at the BS and STARS leads to significant performance degradation due to the beam split issue. To address this, true time delayers (TTDs) are introduced into the conventional hybrid beamforming structure for facilitating wideband beamforming. An iterative algorithm based on the quasi-Newton method is proposed to design the coefficients of the TTDs. Finally, our numerical results confirm the superiority of the STARS over the conventional reconfigurable intelligent surface (RIS). It is also revealed that i) there is only a slight performance loss in terms of SE and EE caused by coupled phase shifts of the STARS in both narrowband and wideband systems, and ii) the conventional hybrid beamforming achieves comparable SE performance and much higher EE performance compared with the full-digital beamforming in narrowband systems but not in wideband systems, where the TTD-based hybrid beamforming is required for mitigating wideband beam split.