The UV-to-IR transitions in p-doped poly(3-hexylthiophene) (P3HT) with alkyl side chains and polar polythiophene with tetraethylene glycol side chains are studied experimentally by means of the absorption spectroscopy and computationally using density functional theory (DFT) and tight-binding DFT. The evolution of electronic structure is calculated as the doping level is varied, while the roles of dopant ions, chain twisting, and pi-pi stacking are also considered, each of these having the effect of broadening the absorption peaks while not significantly changing their positions. The calculated spectra are found to be in good agreement with experimental spectra obtained for the polymers doped with a molybdenum dithiolene complex. As in other DFT studies of doped conjugated polymers, the electronic structure and assignment of optical transitions that emerge are qualitatively different from those obtained through earlier "traditional" approaches. In particular, the two prominent bands seen for the p-doped materials are present for both polarons and bipolarons/polaron pairs. The lowest energy of these transitions is due to excitation from the valence band to a spin-resolved orbitals located in the gap between the bands. The higher-energy band is a superposition of excitation from the valence band to a spin-resolved orbitals in the gap and an excitation between bands.