Auxetic materials, materials demonstrating negative Poisson's ratio, have revolutionized the use of materials in industries, as they demonstrate superb acoustic response, fracture resistance, and energy absorption. For the first time, this study embraces the free vibration of conical shells consisting of an auxetic core with and without ring support under various boundary conditions. First, the material characteristics of the auxetic core are calculated by means of a micromechanical approach. Afterwards, the kinematic motion equations of the conical shell are derived utilizing the first-order shear deformation theory. Finally, the governing equations are solved using the powerful generalized differential quadrature element method (GDQEM). The primary goal of this paper is to study the role of implementing an auxetic core as well as ring support in determining the vibrational behavior of the structure. The results of the study showed that the honeycomb interior angle and the presence of ring support can significantly affect the natural frequency of the structure. Lower frequencies can be reached as the interior angle increases. The importance of ring position is found to be highly dependent on the longitudinal mode shapes of vibration. The impact of ring position on natural frequencies is affected by the semi-vertex angle of the cone, and a shift in frequency peaks can be observed by increasing the semi-vertex angle.