Confined and interface acoustic phonon modes in a cylindrical quantum wire embedded in another material are analytically investigated based on the elastic continuum model by means of the potential theory. Confined acoustic phonon modes are coupled modes of bulk-longitudinal and transverse acoustic waves, classified into torsional, dilatational, and flexural modes due to the rotational symmetry of the modes. Dispersions of the confined modes have subband structures with finite cutoff frequencies owing to quantization of wave vectors in the lateral direction. The density of confined phonon states becomes, as a result, a staircaselike structure. As for the interface modes, regions of material parameters for the possible existence of interface modes are investigated. We found that the existence of interface modes in a quantum wire--surrounding system becomes more sensitive to the combinations of materials than that for a plane interface.