We have monitored the evolution of the transient X-ray source GRO J0422+32 from approximately 2 weeks postdiscovery into its early decline phase at ultraviolet, optical, and radio wavelengths. Optical and ultraviolet spectra exhibit numerous, but relatively weak, high-excitation emission lines such as those arising from He II, N III, N V, and C IV superposed on an intrinsically blue continuum. High-resolution optical spectroscopy reveals line profiles which are double peaked, and in the case of the higher order Balmer lines, superposed on a broad absorption profile. The early outburst optical-ultraviolet continuum energy distribution is well represented by a two power-law fit with a break at approximately equal 4000 A. Radio observations with the Very Large Array (VLA) reveal a flat-spectrum source, slowly increasing in intensity at the earliest epochs observed, followed by an approximate power-law decay light curve with an index of -1. Light curves for each wavelength domain are presented and discussed. Notable are the multiple secondary outbursts seen in the optical more than 1 year postdiscovery, and spectral changes associated with secondary rises seen in the radio and UV. We find that the ultraviolet and optical characteristics of GRO J0422+32 as well as its radio evolution, are similar to other recent well-observed soft X-ray transients (also called X-ray novae) such as Cen X-4, A0620-00 (V616 Mon), and Nova Muscae 1991 (GS 1124-683), suggesting that GRO J0422+32 is also a member of that subclass of low-mass X-ray binaries. We present definitive astrometric determination of the source position, and place an upper limit of R approximately equals 20 from our analysis of the Palomar Observatory Sky Survey (POSS). Additionally, we derive distinct values for color excess from analysis of the optical (E(B-V) = 0.23) and ultraviolet (E(B-V) = 0.4) data, suggesting an intrinsic magnitude of 19-19.5 for the progenitor if it is mid-K dwarf. This leads to a likely range of 2.4-3.0 kpc for the source distance, which is consistent with our separate estimate of 2.4 +/- 0.4 kpc based on measurement of the NaD interstellar line profile. Adopting 2.4 kpc and E(B-V) = 0.23, the outburst absolute magnitude was M approximately equals 0.0, which is a typical value for this class of objects.