An excitonic-insulating system is studied from a viewpoint of the orbital physics in strongly correlated electron systems. An effective model Hamiltonian for low-energy electronic states is derived from the two-orbital Hubbard model with a finite-energy difference corresponding to the crystalline-field splitting. The effective model is represented by the spin operators and the pseudospin operators for the spin-state degrees of freedom. The ground-state phase diagram is analyzed by the mean-field approximation. In addition to the low-spin state and high-spin state phases, two kinds of the excitonic-insulating phases emerge as a consequence of the competition between the crystalline-field effect and the Hund coupling. Transitions to the excitonic phases are classified to an Ising-type transition resulted from a spontaneous breaking of the Z2 symmetry. Magnetic structures in the two excitonic-insulating phases are different from each other: an antiferromagnetic order and a spin nematic order. Collective excitations in each phase are examined using the generalized spin-wave approximation. Characteristics in the Goldstone modes in the excitonic-insulating phases are studied through the calculations of the dynamical correlation functions for the spins and pseudospin operators. Both the transverse and longitudinal spin excitation modes are active in the two excitonic-insulating phases in contrast to the low-spin state and high-spin state phases. Relationships of the present results to the perovskite cobalt oxides are discussed.