Supernova (SN) 2008ax in NGC 4490 was discovered within hours after shock breakout, presenting the rare opportunity to study a core-collapse SN beginning with the initial envelope-cooling phase immediately following shock breakout. We present an extensive sequence of optical and near-infrared spectra, as well as three epochs of optical spectropolarimetry. Our initial spectra, taken two days after shock breakout, are dominated by hydrogen Balmer lines at high velocity. However, by maximum light, He I lines dominated the optical and near-infrared spectra, which closely resembled those of normal Type Ib supernovae (SNe Ib) such as SN 1999ex. This spectroscopic transition defines Type IIb SNe, but the strong similarity of SN 2008ax to normal SNe Ib beginning near maximum light, including an absorption feature near 6270 A due to H Delta *a at high velocities, suggests that many objects classified as SNe Ib in the literature may have ejected similar amounts of hydrogen as SN 2008ax, roughly a few X 0.01 M . Only the unusually early discovery of SN 2008ax allowed us to observe the spectroscopic signatures of the hydrogen-rich outer ejecta. Early-time spectropolarimetry (six and nine days after shock breakout) revealed strong line polarization modulations of 3.4% across H Delta *a, indicating the presence of large asphericities in the outer ejecta and possibly that the spectrum of SN 2008ax could be dependent on the viewing angle. After removal of interstellar polarization, the continuum shares a common polarization angle with the hydrogen, helium, and oxygen lines, while the calcium and iron absorptions are oriented at different angles. This is clear evidence of deviations from axisymmetry even in the outer ejecta. Intrinsic continuum polarization of 0.64% only nine days after shock breakout shows that the outer layers of the ejecta were quite aspherical. A single epoch of late-time spectropolarimetry as well as the shapes of the nebular line profiles demonstrate that asphericities extended from the outermost layers all the way down to the center of this core-collapse SN. SN 2008ax may in some ways be an extragalactic analog of the explosion giving rise to Cassiopeia A, which has recently been determined to be a remnant of an SN IIb.