We present the transmission spectra of the hot Jupiter HD 209458b taken with the Space Telescope Imaging Spectrograph aboard the Hubble Space Telescope. Our analysis combines data at two resolutions and applies a complete pixel-by-pixel limb-darkening correction to fully reveal the spectral line shapes of atmospheric absorption features. Terrestrial-based Na I and H I contamination are identified that mask the strong exoplanetary absorption signature in the Na core, which we find reaches total absorption levels of ~0.11% in a 4.4 Å band. The Na spectral line profile is characterized by a wide absorption profile at the lowest absorption depths and a sharp transition to a narrow absorption profile at higher absorption values. The transmission spectra also show the presence of an additional absorber at ~6250 Å, observed at both medium and low resolutions. We performed various limb-darkening tests, including using high-precision limb-darkening measurements of the Sun to characterize a general trend of ATLAS models to slightly overestimate the amount of limb darkening at all wavelengths, likely due to the limitations of the model's one-dimensional nature. We conclude that, despite these limitations, ATLAS models can still successfully model limb darkening in high signal-to-noise ratio transits of solar-type stars, like HD 209458, to a high level of precision over the entire optical regime (3000-10000 Å) at transit phases between second and third contact.