Measurements are presented of fiducial and total inclusive cross sections for W-plus, W-minus, and Z bosons, decaying to leptonic or muonic final states, in proton-proton collisions at 13 TeV center-of-mass energies, using the ATLAS Detector at CERN's Large Hadron Collider. Data used in these measurements correspond to 81 inverse picobarns of integrated luminosity, recorded during June and July 2015. Cross sections are measured in six independent analysis channels, corresponding to the electronic and muonic final states of W-plus, W-minus, and Z bosons. Electron and muon channels are combined to obtain fiducial and total cross-sections-times-leptonic-branching-ratios for each of W-plus, W-minus, and Z. Lepton universality ratios for W and Z are calculated and found to be compatible with Standard Model expectations and global averages. The ratio of W and Z fiducial cross sections is measured. The ratio of W-plus and W-minus fiducial cross sections is measured. All measurements are compared to NNLO(QCD)+NLO(EW) predictions, based on a selection of modern proton parton distribution functions.