<p>In this paper, we investigate the Davis-Wielandt shell and the numerical range of composition operators on the Hardy space. Firstly, we give a characterization of the Davis-Wielandt shell for multiplication operators with matrix symbols. Subsequently, we characterize the Davis-Wielandt shell of composition operators induced by constant functions, inner functions fixing 0 and elliptic automorphisms of order 2. Furthermore, we analyze the symmetry of the Davis-Wielandt shell for composition operators induced by parabolic automorphisms or elliptic automorphisms. Additionally, we present a complete description of the numerical range for composition operators induced by elliptic automorphisms of order 3.</p>