Efficient processing and transfer of information in neurons have been linked to noise-induced resonance phenomena such as coherence resonance (CR), and adaptive rules in neural networks have been mostly linked to two prevalent mechanisms: spike-timing-dependent plasticity (STDP) and homeostatic structural plasticity (HSP). Thus, this paper investigates CR in small-world and random adaptive networks of Hodgkin-Huxley neurons driven by STDP and HSP. Our numerical study indicates that the degree of CR strongly depends, and in different ways, on the adjusting rate parameter $P$, which controls STDP, on the characteristic rewiring frequency parameter $F$, which controls HSP, and on the parameters of the network topology. In particular, we found two robust behaviors: (i) Decreasing $P$ (which enhances the weakening effect of STDP on synaptic weights) and decreasing $F$ (which slows down the swapping rate of synapses between neurons) always leads to higher degrees of CR in small-world and random networks, provided that the synaptic time delay parameter $τ_c$ has some appropriate values. (ii) Increasing the synaptic time delay $τ_c$ induces multiple CR (MCR) -- the occurrence of multiple peaks in the degree of coherence as $τ_c$ changes -- in small-world and random networks, with MCR becoming more pronounced at smaller values of $P$ and $F$. Our results imply that STDP and HSP can jointly play an essential role in enhancing the time precision of firing necessary for optimal information processing and transfer in neural systems and could thus have applications in designing networks of noisy artificial neural circuits engineered to use CR to optimize information processing and transfer.