This study investigates the rheological properties of dual-network hydrogels based on acrylamide and sodium alginate under large deformations. The concentration of calcium ions affects the nonlinear behavior, and all gel samples exhibit strain hardening, shear thickening, and shear densification. The paper focuses on systematic variation of the alginate concentration—which serves as second network building blocks—and the Ca2+-concentration—which shows how strongly they are connected. The precursor solutions show a typical viscoelastic solution behavior depending on alginate content and pH. The gels are highly elastic solids with only relatively small viscoelastic components, i.e., their creep and creep recovery behavior are indicative of the solid state after only a very short time while the linear viscoelastic phase angles are very small. The onset of the nonlinear regime decreases significantly when closing the second network (alginate) upon adding Ca2+, while at the same time the nonlinearity parameters (Q0, I3/I1, S, T, e3/e1, and v3/v1) increase significantly. Further, the tensile properties are significantly improved by closing the alginate network by Ca2+at intermediate concentrations.