We study an interacting electron gas in a quantum wire within the Hartree-Fock random phase approximation. Vertex corrections to the electron spin polarizability due to the electronic exchange interaction are important giving rise to spin-density excitations (SDE) with large oscillator strength shifted to lower energies with respect to single-particle states. The energy of intersubband SDE oscillates with the number of subbands occupied and has a minimum when a subband energy is close to the chemical potential. Intrasubband SDE have a linear dispersion at small wave-vectors. The corresponding sound velocity is reduced with respect to the Fermi velocity due to exchange interaction within the occupied subbands and exchange screening caused by virtual transitions to upper subbands. For intersubband and intrasubband charge-density excitations (CDE) vertex corrections are of less importance. For only a single subband occupied the screening of CDE and SDE in the Tomonaga-Luttinger model due to virtual transitions to upper subband is studied, where the virtual transitions are treated within the Hartree-Fock approximation. The calculations are in good qualitative agreement with experiments. Comment: 29 pages, 8 figures, RevTex. To appear in Phys. Rev. B