This work aimed to determine the effect of the addition of different types of reinforcing fibers on the strength properties of geopolymers such as flexural and compressive strength. Geopolymers are an attractive alternative to conventional binders and building materials; however, one of the main problems of their widespread use is their low resistance to brittle fracture. To improve the mechanical properties, reinforcement in the form of glass, carbon, and basalt fibers (as grids) was applied to geopolymers in the following work. Additionally, composites with these fibers were produced not only in the matrix of pure geopolymer but also as a hybrid variant with the addition of cement. Furthermore, basalt grids were used as reinforcement for geopolymers not only based on ash but also metakaolin. An additional variable used in the study was the molar concentration of the alkali solution (5 M and 10 M) for the different types of geopolymer samples. The mechanical properties of geopolymer materials and geopolymer–cement hybrids are the highest when reinforcement in the form of carbon fiber is used. Strength values for geopolymers reinforced with basalt mats depend on the number of reinforcement layers and the concentration of the alkaline solution used. All produced composites were tested for compressive strength and bending strength. When using basalt mesh, it was possible to achieve a bending strength of 12 MPa. The highest compressive strength that was achieved was the value of 66 MPa, while for samples not reinforced with fibers, only about 40 MPa was achieved.