It is widely known that glass–epoxy laminates are renowned for their high stiffness, good thermal properties, and economic qualities. For this reason, composite materials find successful applications in various industrial sectors such as aerospace, astronautics, the storage sector, and energy. The aim of this study was to investigate the mechanical and thermal properties of composite materials comprising two different types of epoxy resin and three different hardeners, both at room temperature and under cryogenic conditions. The samples were produced at IZOERG (Gliwice, Poland) using a laboratory hot-hydraulic-press technique. During cyclic loading–unloading tests, degradation up to a strain level of 0.6% was observed both at room temperature (RT) and at 77 K. For a glass-reinforced composite with YDPN resin (EP_1_1), the highest degradation was recorded at 18.84% at RT and 33.63% at 77 K. We have also investigated the temperature dependence of thermal conductivity for all samples in a wide temperature range down to 5 K. The thermal conductivity was found to be low and had a relative difference of up to 20% among the composites. The experimental results indicated that composites under cryogenic conditions exhibited less damage and were stiffer. It was confirmed that the choice of hardener significantly influenced both properties.