Fiber reinforcement is currently most often used in floors, railway sleepers, prefabricated structural elements such as slabs, beams and tanks, and in small architecture elements. Designing elements or structures made of fiber-reinforced concrete requires knowledge of its basic mechanical parameters. In the case of concretes with metallic fibers, the literature can find many tests and standard guidelines regarding compressive, flexural, tensile strength and fracture energy. The properties of concretes with non-metallic fibers are slightly less recognized, especially concretes with new types of polymer fibers. Additionally, the lack of standardized methods of testing concrete with polymer fibers make their application much more difficult. In the article, the possibility of using the EN 14651 standard to assess the flexural tensile strength of concrete with the addition of 2.0 and 3.0 kg/m3 of synthetic fibers with different geometry and form was presented. There was a 5.5–13.5% increase in the flexural tensile strength depending on the mixture type. Moreover, in the case of fiber-reinforced concretes, the ductility was enhanced and the samples were characterized by significant residual flexural tensile strengths. Additionally, from the workability tests it was concluded that after the incorporation of fibers, the consistency class decreased by one, two or three. Nevertheless, the compressive strengths of concrete with and without fibers were very similar to each other, and varied from 58.05 to 61.31 MPa. Moreover, it was concluded that results obtained from three-point bending tests significantly differed from empirical formulas for the calculation of the flexural tensile strength of fiber-reinforced concretes with dispersed steel fibers present in the literature. As a result, the new formula determined by the authors was proposed for concrete with polymer fibers with a nominal fiber content ≤1.0% and slenderness of up to 200. It must be mentioned that the formula gave a very good agreement with studies presented in different literature positions. In addition, an attempt was made to evaluate the strengths of tested mixes in accordance with the Model Code 2010. However, it occurred that the proposed fiber-reinforced concrete mixtures would not be able to replace traditional reinforcement in a form of steel bars. Furthermore, in uniaxial tensile tests, it was not possible to determine the σ–w graphs, and received results for maximum tensile strength did not show the clear influence of fibers incorporation on concrete. Then, the fracture energy enhancement (from about 16 to 22 times) and dependencies: crack mouth opening displacement–deflection; crack mouth opening displacement–crack tip opening displacement; and crack tip opening displacement–deflection were analyzed. Finally, the results from flexural tensile tests were compared with measurements of the surface displacement field obtained through the Digital Image Correlation technique. It was concluded that this technique can be successfully used to determine the crack mouth and crack tip opening displacements with very high accuracy.