The STAR Collaboration reports measurements of the transverse single-spin asymmetry (TSSA) of inclusive pi(0) at center-of-mass energies (root s) of 200 GeV and 500 GeV in transversely polarized proton-proton collisions in the pseudo-rapidity region 2.7 to 4.0. The results at the two different energies show a continuous increase of the TSSA with Feynman-x, and, when compared to previous measurements, no dependence on root s from 19.4 GeV to 500 GeV is found. To investigate the underlying physics leading to this large TSSA, different topologies have been studied. pi(0) with no nearby particles tend to have a higher TSSA than inclusive pi(0). The TSSA for inclusive electromagnetic jets, sensitive to the Sivers effect in the initial state, is substantially smaller, but shows the same behavior as the inclusive pi(0) asymmetry as a function of Feynman-x. To investigate final-state effects, the Collins asymmetry of pi(0) inside electromagnetic jets has been measured. The Collins asymmetry is analyzed for its dependence on the pi(0) momentum transverse to the jet thrust axis and its dependence on the fraction of jet energy carried by the pi(0). The asymmetry was found to be small in each case for both center-of-mass energies. All the measurements are compared to QCD-based theoretical calculations for transverse-momentum-dependent parton distribution functions and fragmentation functions. Some discrepancies are found, which indicates new mechanisms might be involved.