Two-dimensional materials (e.g., graphene and transition metal dichalcogenides) and their heterostructures have enormous applications in electrochemical energy storage systems such as batteries. A comprehensive and solid understanding of these materials' thermal transport and mechanism is essential for practical device design. Several advanced experimental techniques have been developed to measure the intrinsic thermal conductivity of materials. However, experiments have challenges in providing improved control and characterization of complex structures, especially for low-dimensional materials. Theoretical and simulation tools, such as first-principles calculations, Boltzmann transport equations, molecular dynamics simulations, lattice dynamics simulation, and nonequilibrium Green's function, provide reliable predictions of thermal conductivity and physical insights to understand the underlying thermal transport mechanism in materials. However, doing these calculations requires high computational resources. The development of new materials synthesis technology and fast-growing demand for rapid and accurate prediction of physical properties requires novel computational approaches. The machine learning method provides a promising solution to address such needs. This review details the recent development in atomistic/molecular studies and machine learning of thermal transport in two-dimensional materials. The paper also addresses the latest significant experimental advances. However, designing the best two-dimensional materials-based heterostructures is like a multivariate optimization problem. For example, a particular heterostructure may be suitable for thermal transport but can have lower mechanical strength/stability. For bilayer and multilayer structures, the interlayer distance may influence the thermal transport properties and interlayer strength. Therefore, the last part of this review addresses the future research direction in two-dimensional materials-based heterostructure design for thermal transport in energy storage systems.