Octupole deformations and related collective excitations are studied using the framework of nuclear density functional theory. Axially-symmetric quadrupole-octupole constrained self-consistent mean-field (SCMF) calculations with a choice of universal energy density functional are performed for those nuclei such as light actinides, Xe, Ba, Ce, and Nd isotopes from proton-rich to neutron-rich sides, and neutron-rich Ge, Se, and Kr, in which enhanced octupole correlations are expected to occur. Low-energy positive- and negative-parity spectra and transition strengths are computed in terms of the interacting boson model, with the parameters determined by the constrained SCMF calculations. Octupole-deformed equilibrium states are found in the potential energy surfaces of the nuclei in the regions corresponding to the neutron or/and proton numbers equal to 34, 56, 88, and 134. The evolution of the calculated spectroscopic properties indicate the onset of octupole deformation and exhibit signatures of octupole shape-phase transitions around these nucleon numbers.