We study spontaneous dimerization and emergent criticality in a spin-3/2 chain with antiferromagnetic nearestneighbor J(1), next-nearest-neighbor J(2), and three-site J(3) interactions. In the absence of three-site interaction J(3), we provide evidence that the model undergoes a remarkable sequence of three phase transitions as a function of J(2)/J(1), going successively through a critical-commensurate phase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommensurate order, to end up in a fully dimerized phase at very large J(2)/J(1). In the field-theory language, this implies that the coupling constant of the marginal operator responsible for dimerization changes sign three times. For large enough J(3), the fully dimerized phase is stabilized for all J(2), and the phase transitions between the critical phases and this phase are both Wess-Zumino-Witten (WZW) SU(2)(3) along part of the boundary and turn first order at some point due to the presence of a marginal operator in the WZW SU(2) 3 model. By contrast, the transition between the two-dimerized phase is always first order, and the phase transitions between the partially dimerized phase and the critical phases are Kosterlitz-Thouless. Finally, we discuss the intriguing spin-1/2 edge states that emerge in the partially dimerized phase for even chains. Unlike their counterparts in the spin-1 chain, they are not confined and disappear upon increasing J(2) in favor of a reorganization of the dimerization pattern.