We study charged particle production (pT>0.5 GeV/c, |η|<0.8) in proton-antiproton collisions at total center-of-mass energies s=300 GeV, 900 GeV, and 1.96 TeV. We use the direction of the charged particle with the largest transverse momentum in each event to define three regions of η-φ space: "toward", "away", and "transverse." The average number and the average scalar pT sum of charged particles in the transverse region are sensitive to the modeling of the "underlying event." The transverse region is divided into a MAX and MIN transverse region, which helps separate the "hard component" (initial and final-state radiation) from the "beam-beam remnant" and multiple parton interaction components of the scattering. The center-of-mass energy dependence of the various components of the event is studied in detail. The data presented here can be used to constrain and improve QCD Monte Carlo models, resulting in more precise predictions at the LHC energies of 13 and 14 TeV.