We carry out a detailed analysis of the light Higgs bosons in supersymmetric left-right models (SLRM). This includes models with minimal particle content and those with additional Higgs superfields. We also consider models with non-renormalizable higher-dimensional terms. We obtain an upper bound on the mass of the lightest CP-even neutral Higgs boson in these models. The upper bound depends only on the gauge couplings, and the vacuum expectation values of those neutral Higgs fields which control the spontaneous breakdown of the $SU(2)_L \times U(1)_Y$ gauge symmetry. We calculate the one-loop radiative corrections to this upper bound, and evaluate it numerically in the minimal version of the supersymmetric left-right model. We consider the couplings of this lightest CP-even Higgs boson to the fermions, and show that in a phenomenologically viable model the branching ratios are similar to the corresponding branching ratios in the minimal supersymmetric standard model (MSSM). We then study the most promising particle for distinguishing the SLRM from other models, namely the doubly charged Higgs boson. We obtain the mass of this doubly charged Higgs boson in different types of supersymmetric left-right models, and discuss its phenomenology.