We investigate the thermal evolution of magnetic correlations of the Co-based honeycomb lattice ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ with $^{23}\mathrm{Na}$ nuclear magnetic resonance and static magnetic susceptibility $\ensuremath{\chi}(T)$. The studied compound shows three-dimensional (3D) long-range magnetic ordering at ${T}_{\mathrm{N}}=26$ K. On cooling through ${T}^{*}\ensuremath{\approx}110$ K, a simple paramagnetic state undergoes a crossover to a correlated paramagnetic state featuring a power-law dependence of the nuclear spin-lattice $(1/{T}_{1})$ and spin-spin $(1/{T}_{2})$ relaxation rates as well as of the out-of-plane $\ensuremath{\chi}(T)$. The magnetic-field-direction dependence of $1/{T}_{1}$, $1/{T}_{2}$, and $\ensuremath{\chi}(T)$ uncovers anisotropic spin-spin correlations of a two-dimensional (2D) renormalized classical character. In a magnetically ordered state, we are able to identify four successive transitions or crossovers occurring at ${T}_{\mathrm{N}}=26$, ${T}_{\mathrm{N}1}=16$, ${T}_{\mathrm{N}2}=7$, and ${T}_{\mathrm{N}3}=3.5$ K. The multiple transitions and crossovers are associated with the coexistence of 2D and 3D magnetic orders or reorientation of the ordered spins. Our results suggest the presence of various types of frustrating interactions and their energy hierarchy that control complex magnetic structures and anisotropic magnetism.