Hydrogels as biomaterials possess appropriate physicochemical and mechanical properties that enable the formation of a three-dimensional, stable structure used in tissue engineering and 3D printing. The integrity of the hydrogel composition is due to the presence of covalent or noncovalent cross-linking bonds. Using various cross-linking methods and agents is crucial for adjusting the properties of the hydrogel to specific biomedical applications, e.g., for direct bioprinting. The research subject was mixtures of gel-forming polymers: sodium alginate and gelatin. The polymers were cross-linked ionically with the addition of CaCl2 solutions of various concentrations (10%, 5%, 2.5%, and 1%) and covalently using squaric acid (SQ) and dialdehyde starch (DAS). Initially, the polymer mixture's composition and the hydrogel cross-linking procedure were determined. The obtained materials were characterized by mechanical property tests, swelling degree, FTIR, SEM, thermal analysis, and biological research. It was found that the tensile strength of hydrogels cross-linked with 1% and 2.5% CaCl2 solutions was higher than after using a 10% solution (130 kPa and 80 kPa, respectively), and at the same time, the elongation at break increased (to 75%), and the stiffness decreased (Young Modulus is 169 kPa and 104 kPa, respectively). Moreover, lowering the concentration of the CaCl2 solution from 10% to 1% reduced the final material's toxicity. The hydrogels cross-linked with 1% CaCl2 showed lower degradation temperatures and higher weight losses than those cross-linked with 2.5% CaCl2 and therefore were less thermally stable. Additional cross-linking using SQ and DAS had only a minor effect on the strength of the hydrogels, but especially the use of 1% DAS increased the material's elasticity. All tested hydrogels possess a 3D porous structure, with pores of irregular shape and heterogenic size, and their swelling degree initially increased sharply to the value of approx. 1000% during the first 6 h, and finally, it stabilized at a level of 1200-1600% after 24 h. The viscosity of 6% gelatin and 2% alginate solutions with and without cross-linking agents was similar, and they were only slightly shear-thinning. It was concluded that a mixture containing 2% sodium alginate and 6% gelatin presented optimal properties after gel formation and lowering the concentration of the CaCl2 solution to 1% improved the hydrogel's biocompatibility and positively influenced the cross-linking efficiency. Moreover, chemical cross-linking by DAS or SQ additionally improved the final hydrogel's properties and the mixture's printability. In conclusion, among the tested systems, the cross-linking of 6% gelatin-2% alginate mixtures by 1% DAS addition and 1% CaCl2 solution is optimal for tissue engineering applications and potentially suitable for 3D printing.