Many applications in biomedicine and synthetic bioengineering depend on the ability to understand, map, predict, and control the complex, context-sensitive behavior of chemical and genetic networks. The emerging field of diverse intelligence has offered frameworks with which to investigate and exploit surprising problem-solving capacities of unconventional agents. However, for systems that are not conventional animals used in behavior science, there are few quantitative tools that facilitate exploration of their competencies, especially when their complexity makes it infeasible to use unguided exploration. Here, we formalize and investigate a view of gene regulatory networks as agents navigating a problem space. We develop automated tools to efficiently map the repertoire of robust goal states that GRNs can reach despite perturbations. These tools rely on two main contributions that we make in this paper: (1) Using curiosity-driven exploration algorithms, originating from the AI community to explore the range of behavioral abilities of a given system, that we adapt and leverage to automatically discover the range of reachable goal states of GRNs and (2) Proposing a battery of empirical tests inspired by implementation-agnostic behaviorist approaches to assess their navigation competencies. Our data reveal that models inferred from real biological data can reach a surprisingly wide spectrum of steady states, while showcasing various competencies that living agents often exhibit, in physiological network dynamics and that do not require structural changes of network properties or connectivity. Furthermore, we investigate the applicability of the discovered "behavioral catalogs" for comparing the evolved competencies across classes of evolved biological networks, as well as for the design of drug interventions in biomedical contexts or for the design of synthetic gene networks in bioengineering. Altogether, these automated tools and the resulting emphasis on behavior-shaping and exploitation of innate competencies open the path to better interrogation platforms for exploring the complex behavior of biological networks in an efficient and cost-effective manner.