In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we prove that for an Alexander quandle $A_t(G)$ over a finite abelian group $G$, the connected components of the Cayley graph correspond to the cosets of the subgroup $\mathrm{im}(\mathrm{id}-t)$. We also show that the Cayley graphs of generalized Alexander quandles are regular. When the defining automorphism is inner, we give an explicit description of the forward orbits and prove that the connected components correspond to cosets of the subgroup generated by commutators with the defining element.