A surface model on compartmentalized spheres is studied by using the Monte Carlo simulation technique with dynamical triangulations. We found that the model exhibits a variety of phases: The spherical phase, the tubular phase, the planar phase, the wormlike planar phase, the wormlike long phase, the wormlike short phase, and the collapsed phase. We also demonstrated that almost all phases are separated from their neighboring phases by first-order transitions. Mechanical strength of the surface is given only by elastic skeletons, which are the compartment boundaries, and therefore vertices diffuse freely inside the compartments. We confirm that the cytoskeletal structure and the lateral diffusion of vertices are origins of such a variety of phases.