[Abridged] We present new optical integral field spectroscopy (Gemini South) and submillimeter spectroscopy (Submillimeter Array) of the central galaxy in the Phoenix cluster (SPT-CLJ2344-4243). This cluster was previously reported to have a massive starburst (~800 Msun/yr) in the central, brightest cluster galaxy, most likely fueled by the rapidly-cooling intracluster medium. These new data reveal a complex emission-line nebula, extending for >30 kpc from the central galaxy. The total Halpha luminosity, assuming Halpha/Hbeta = 2.85, is L_Ha = 7.6 +/- 0.4 x10^43 erg/s, making this the most luminous emission line nebula detected in the center of a cool core cluster. Overall, the relative fluxes of the low-ionization lines (e.g., [O II], Hbeta) to the UV continuum are consistent with photoionization by young stars. In both the center of the galaxy and in a newly-discovered highly-ionized plume to the north of the galaxy, the ionization ratios are consistent with both shocks and AGN photoionization. We speculate that this extended plume may be a galactic wind, driven and partially photoionized by both the starburst and central AGN. We find evidence for shocks throughout the ISM of the central galaxy, most likely driven by a combination of stellar winds from massive young stars, core-collapse supernovae, and the central AGN. In addition to the warm, ionized gas, we detect a substantial amount of cold, molecular gas via the CO(3-2) transition, coincident in position with the galaxy center. We infer a molecular gas mass of M_H2 = 2.2 +/- 0.6 x10^10 Msun, which implies that the starburst will consume its fuel in ~30 Myr if it is not replenished. The combination of the high level of turbulence in the warm phase and the high L_IR/M_H2 ratio suggests that this violent starburst may be in the process of quenching itself.