Hydrogels tested and evaluated in this study were developed for the possibility of their use as the bioinks for 3D direct bioprinting. Procedures for preparation and sterilization of hydrogels and the speed of the bioprinting were developed. Sodium alginate gelatine hydrogels were characterized in terms of printability, mechanical, and biological properties (viability, proliferation ability, biocompatibility). A hydrogel with the best properties was selected to carry out direct bioprinting tests in order to determine the parameters of the bioink, adapted to print with use of the designed and constructed bioprinter and provide the best conditions for cell growth. The obtained results showed the ability to control mechanical properties, biological response, and degradation rate of hydrogels through the use of various solvents. The use of a dedicated culture medium as a solvent for the preparation of a bioink, containing the predicted cell line, increases the proliferation of these cells. Modification of the percentage of individual components of the hydrogel gives the possibility of a controlled degradation process, which, in the case of printing of temporary medical devices, is a very important parameter for the hydrogels’ usage possibility—both in terms of tissue engineering and printing of tissue elements replacement, implants, and organs.