Additive manufacturing, through the process of thermoplastic extrusion of filament, allows the manufacture of complex composite sandwich structures in a short time with low costs. This paper presents the design and fabrication by Fused Filament Fabrication (FFF) of composite sandwich structures with short fibers, having three core types C, Z, and H, followed by mechanical performance testing of the structures for compression and bending in three points. Flatwise compression tests and three-point bending have clearly indicated the superior performance of H-core sandwich structures due to dense core structures. The main modes of failure of composite sandwich structures were analyzed microscopically, highlighting core shear buckling in compression tests and face indentation in three-point bending tests. The strength–mass ratio allowed the identification of the structures with the best performances considering the desire to reduce the mass, so: the H-core sandwich structures showed the best results in compression tests and the C-core sandwich structures in three-point bending tests. The feasibility of the FFF process and the three-point bending test of composite wing sections, which will be used on an unmanned aircraft, have also been demonstrated. The finite element analysis showed the distribution of equivalent stresses and reaction forces for the composite wing sections tested for bending, proving to validate the experimental results.