The governing linear equilibrium equations and their corresponding boundary conditions are applied to analyze both the mechanical and thermal stability of sandwich plates and shells. These structures feature anisotropic laminated composite facings that are symmetric about both their local and global mid-surfaces. For the mechanical buckling analysis, a doubly curved sandwich panel with a weak core, supported by two types of simply supported boundary conditions, is considered. For thermal buckling, a flat sandwich panel with a strong core, subjected to a uniform temperature rise, is analyzed. Simply supported boundary conditions are assumed, with both normal and tangential displacements fixed along all four edges. The governing approximate solutions for both mechanical and thermal buckling are derived using the Extended (Modified) Galerkin Method and the Classical Galerkin Method, respectively. The effects of various structural configurations, such as the tailoring of face sheets, panel thickness, aspect ratio, and others, on the critical loading of the sandwich panel are examined. Finally, the results are validated against existing literature.