Neutrino oscillation experiments are designed to measure neutrino masses and mixing parameters by scattering them off nuclei such as carbon, oxygen, and argon in detectors. Predictions of neutrino-nuclei cross sections from the Standard Model are needed to extract these parameters, but their theoretical uncertainties remain large due to the complexity ofnuclear and hadronic physics. This situation needs to be improved in order to satisfy the precision needs of future experiments. In this dissertation, we focus on working towards a first-principles calculation of the nucleon axial form factor with lattice quantum chromodynamics (QCD). Nucleon axial form factor, which parametrizes the weak responses of a proton or neutron, is difficult to measure experimentally, and it is a dominant uncertainty in neutrino-nuclei cross-section calculationsfor incoming neutrino energies at around 1 GeV. So a theoretical calculation with lattice QCD provides a non-ambiguous determination of the form factor that could help reducing the uncertainty. The notorious signal-to-noise problem renders calculations of nucleon observables computationally intensive in lattice QCD. In this work, we investigate the use of staggered fermions in nucleon calculations. Staggered fermions are the most computationally efficient fermion discretization in lattice field theory, but certain theoretical issues have so far prevented their applications to nucleon physics. As a stepping stone towards a full calculation of the nucleon axial form factor, this dissertation provides a comprehensive theoretical framework on how to calculate the nucleon mass, vector charge, and axial charge with staggered fermions, together with numerical results demonstrating the methodology. This framework can be generalized to the form factor calculation that will appear in the near future.