An analysis has been presented of the effect of elastic foundation and uniform in-plane peripheral loading on the natural frequencies and mode shapes of circular plates of varying thickness exhibiting bi-directional functionally graded characteristics, on the basis of first order shear deformation theory. The material properties of the plate are varying following a power-law in both the radial and transverse directions. The numerical solutions of the coupled differential equations leading the motion of simply supported and clamped plates acquired by using Hamilton’s principle, is attained by harmonic differential quadrature method. The effect of different plate parameters namely gradient index, heterogeneity parameter, density parameter, taper parameter and thickness parameter is illustrated on the vibration characteristics for the first three modes of vibration for various values of in-plane peripheral loading parameter together with foundation parameter. Critical buckling loads in compression are calculated for both the boundary conditions by putting the frequencies to zero. The reliability of the present technique is confirmed by comparing the results with exact values and results of published work.