Different contact materials along with optimization of the deposition techniques expand the possibilities to obtain high performance room temperature CdTe-based X/γ-ray detectors. The heterostructures with ohmic (MoOx) and Schottky (MoOx, TiOx, TiN, and In) contacts, created by DC reactive magnetron sputtering and vacuum thermal evaporation, as well as In/CdTe/Au diodes with a p-n junction, formed by laser-induced doping, were developed and investigated. Depending on the pre-treatment of the surface of semi-insulating p-CdTe crystals, deposition of a MoOx film formed an ohmic or Schottky contact. Based on the calculations and IV characteristics of the fabricated Mo-MoOx/р-CdTe/MoOx-Mo, In/CdTe/MoOx-Mo, Ti-TiOx/р-CdTe/MoOx-Mo, and Ti-TiN/р-CdTe/MoOx-Mo Schottky-diode detectors, the current transport processes were described in the models of the carrier generation-recombination within the space-charge region (SCR) at low bias voltages, and space-charge limited current at higher voltages, respectively. The energies of generation–recombination centers, density of trapping centers, and carrier lifetimes were determined. A sharp increase in the reverse current in the Mo-MoOx/р-CdTe/MoOx-Mo, Ti-TiOx/р-CdTe/MoOx-Mo, and Ti-TiN/р-CdTe/MoOx-Mo heterostructures at high bias was discussed in frames of the Poole–Frenkel emission model. Nanosecond laser irradiation of the In electrode, pre-deposited on the p-CdTe crystals, resulted in an increase in the voltage range, corresponding to the carrier generation-recombination in the SCR, in the IV characteristics of the In/CdTe/Au diodes. Such In/CdTe/Au p-n junction diodes demonstrated high energy resolutions (7%@59.5 keV, 4%@122 keV, and 1.6%@662 keV).