Giant resonances (GRs) provide crucial insights into nuclear physics and astrophysics. Exciting GRs using particles like electrons is effective, yet the angular momentum (AM) transfer of electrons, including both intrinsic spin and orbital degrees of freedom in inelastic scattering, has never been studied. Here, we investigate AM transfer in GRs excited by plane-wave and vortex electrons, developing a comprehensive AM-resolved inelastic electron scattering theory. We find that even plane-wave electrons can model-independently extract transition strengths of higher multipolarity by selecting specific AM states of scattered electrons. Additionally, relativistic vortex electrons with an orbital angular momentum of ±1 can be efficiently generated. Vortex electrons can also be used to extract GR transition strength as in the plane-wave case, regardless of the position of the nucleus relative to the beam axis. Furthermore, relativistic vortex electrons with larger orbital angular momentum can be generated for on-axis nuclei due to AM conservation. Our method offers new perspectives for nuclear structure research and paves the way for generating vortex particles.