The dimensionless fundamental frequencies and critical axial loads of sandwich cylindrical thin shell with a functionally graded (FG) core are studied by taking into account the combined and separately influences of the shear stresses and rotary inertia. The governing equations of sandwich cylindrical shell with an FG core are derived based on Donnell's shell theory using the shear deformation theory. The governing equations are reduced the sixth-order algebraic equation using the Galerkin's method. Numerically solving this algebraic equation gives the magnitudes of the dimensionless fundamental frequency. In addition, the expressions for the dimensionless fundamental frequencies and critical axial loads of the sandwich cylindrical shell containing an FG core with and without the shear stresses are obtained in a special case. To validate the present method, the numerical example is presented and compared with the available existing results. Finally, the influences of variations of the FG core, shear stresses, rotary inertia and sandwich shell geometry parameters on the dimensionless fundamental frequencies and critical axial loads are analyzed numerically.