The chemical stability and adsorptive/catalytic properties of the most widely studied metal–organic framework (MOF), which is HKUST-1, can be improved by its combination with graphene oxide (GO) or reduced graphene oxide (rGO). The chemistry of GO or rGO surfaces has a significant impact on their interaction with MOFs. In this work, we demonstrate that GO and rGO interaction with HKUST-1 influences the morphology and textural properties but has no impact on the thermal stability of the final composites. We also show that synthesis environment, e.g., stirring, to some extent influences the formation of HKUST-1/GO and HKUST-1/rGO composites. Homogeneous samples of the sandwich-type composite can be obtained when using reduced graphene oxide decorated with copper (Cu/rGO), which, owing to the presence of Cu sites, allows the direct crystallisation of HKUST-1 and its further growth on the graphene surface. This work is the first part of our research on HKUST-1/GO and HKUST-1/rGO and deals with the influence of the type of graphene material and synthesis parameters on the composites’ physicochemical properties that were determined by using X-ray diffraction, scanning and transmission electron microscopy, Fourier transform infrared spectroscopy, X-ray photoelectron spectroscopy, and thermogravimetric analysis.