In this article, we continue the program started in [2] of exploring an important class of thermodynamic systems from a geometric point of view. The contents of this paper and the one already published in [2] provide a geometrical formulation, which tries to shed more light on the properties of thermodynamic systems without claiming to be a definitive theory. In order to model the time evolution of systems verifying the two laws of thermodynamics, we show that the notion of evolution vector field is adequate to appropriately describe such systems. Our formulation naturally arises from the introduction of a skew-symmetric bracket to which numerical methods based on discrete gradients fit nicely. Moreover, we study the corresponding Lagrangian and Hamiltonian formalism, discussing the fundamental principles from which the equations are derived. An important class of systems that is naturally covered by our formalism are composed thermodynamic systems, which are described by at least two thermal variables and exchange heat between its components.