The present study examines the formulation of a biocompatible hydrogel bioink for 3D bioprinting, integrating poly(ethylene glycol) diacrylate (PEGDA) and sodium alginate (SA) using a double-network approach. These materials were chosen for their synergistic qualities, with PEGDA contributing to mechanical integrity and SA ensuring biocompatibility. Fibroblast cells were included in the bioink and printed with a Reg4Life bioprinter employing micro-extrusion technology. The optimisation of printing parameters included needle size and flow velocities. This led to precise structure development and yielded results with a negligible deviation in printed angles and better control of line widths. The rheological characteristics of the bioink were evaluated, demonstrating appropriate viscosity and shear-thinning behaviour for efficient extrusion. The mechanical characterisation revealed an average compressive modulus of 0.38 MPa, suitable for tissue engineering applications. The printability of the bioink was further confirmed through the evaluations of morphology and diffusion rates, confirming structural integrity. Biocompatibility assessments demonstrated a high cell viability rate of 82.65% following 48 h of incubation, supporting the bioink’s suitability for facilitating cell survival. This study introduced a reliable technique for producing tissue-engineered scaffolds that exhibit outstanding mechanical characteristics and cell viability, highlighting the promise of PEGDA–SA hydrogels in bioprinting applications.