We report scanning tunneling microscopy and scanning tunneling spectroscopy (STS) measurements of monolayer and bilayer WSe_{2}. We measure a band gap of 2.21±0.08 eV in monolayer WSe_{2}, which is much larger than the energy of the photoluminescence peak, indicating a large excitonic binding energy. We additionally observe significant electronic scattering arising from atomic-scale defects. Using Fourier transform STS, we map the energy versus momentum dispersion relations for monolayer and bilayer WSe_{2}. Further, by tracking allowed and forbidden scattering channels as a function of energy we infer the spin texture of both the conduction and valence bands. We observe a large spin-splitting of the valence band due to strong spin-orbit coupling, and additionally observe spin-valley-layer coupling in the conduction band of bilayer WSe_{2}.