Observations show that waves are ubiquitous in the solar atmosphere and may play an important role for plasma heating. The study of waves in the solar corona is usually based on linear ideal magnetohydrodynamics (MHD) for a fully ionized plasma. However, the plasma in the photosphere and the chromosphere is only partially ionized. Here we theoretically investigate the impact of partial ionization on MHD wave propagation in cylindrical flux tubes in a two-fluid model. We derive the general dispersion relation that takes into account the effects of neutral-ion collisions and the neutral gas pressure. We assumed the neutral-ion collision frequency to be an arbitrary parameter. Specific results for transverse kink modes and slow magnetoacoustic modes are shown. We find that the wave frequencies only depend on the properties of the ionized fluid when the neutral-ion collision frequency is much lower that the wave frequency. For high collision frequencies that realistically represent the solar atmosphere, ions and neutrals behave as a single fluid with an effective density corresponding to the sum of densities of fluids plus an effective sound velocity computed as the average of the sound velocities of ions and neutrals. The MHD wave frequencies are modified accordingly. The neutral gas pressure can be neglected when studying transverse kink waves but it has to be included for a consistent description of slow magnetoacoustic waves. The MHD waves are damped by neutral-ion collisions. The damping is most efficient when the wave frequency and the collision frequency are on the same order of magnitude. For high collision frequencies slow magnetoacoustic waves are more efficiently damped than transverse kink waves. In addition, we find the presence of cut-offs for certain combinations of parameters that cause the waves to become non-propagating.