We present and discuss the complete set of infrared-active phonon modes in monoclinic-symmetry crystal modification gallium oxide (gallia, β-Ga2O3). The phonon mode set is obtained from a comprehensive analysis of generalized spectroscopic ellipsometry data in the farinfrared and infrared spectral regions investigating various n-type electrically conductive single crystal samples with different free electron volume density parameters cut under different orientations. The analysis of the ellipsometry data is performed using an eigendielectric displacement vector summation (EDVS) approach. In this approach, the effect of the free charge carriers onto the lattice modes of intrinsic β-Ga2O3 are removed by calculation. Density functional theory calculations are performed in the general gradient approximation and all phonon modes at the Brillouin-center and their displacement direction dependencies are obtained. Transverse and longitudinal optical phonon mode parameters polarized within the monoclinic plane as well as perpendicular to the monoclinic plane agree excellently between experiment and theory. We also present and discuss the directional limiting frequency parameters within the monoclinic plane, the shape and anisotropy of the reststrahlen band, and the order of the phonon modes in semiconductors with polar phonon modes and monoclinic crystal structure. We further present and discuss the effect of coupling of longitudinal optical phonons with free charge carriers, leading to longitudinal-phonon-plasmon coupled modes. We reveal that the coupled modes, which affect electric and thermal transport, change amplitude, frequency, and direction within the monoclinic plane as a function of free electron concentration. Finally, we show optical Hall effect measurements, and provide experimentally determined effective electron mass parameters in β-Ga2O3 for moderately-doped n-type samples.