We present constraints on the nature of the first galaxies selected at 350 mu m. The sample includes galaxies discovered in the deepest blank-field survey at 350 mu m (in the Bootes Deep Field) and also later serendipitous detections in the Lockman Hole. In determining multiwavelength identifications, the 350 mu m position and map resolution of the second generation Submillimeter High Angular Resolution Camera are critical, especially in the cases where multiple radio sources exist and the 24 mu m counterparts are unresolved. Spectral energy distribution templates are fitted to identified counterparts, and the sample is found to comprise IR-luminous galaxies at 1 < z < 3 predominantly powered by star formation. The first spectrum of a 350 mu m selected galaxy provides an additional confirmation, showing prominent dust grain features typically associated with star-forming galaxies. Compared to submillimeter galaxies selected at 850 and 1100 mu m, galaxies selected at 350 mu m have a similar range of far-infrared color temperatures. However, no 350 mu m selected sources are reliably detected at 850 or 1100 mu m. Galaxies in our sample with redshifts 1 < z < 2 show a tight correlation between the far-and mid-infrared flux densities, but galaxies at higher redshifts show a large dispersion in their mid-to far-infrared colors. This implies a limit to which the mid-IR emission traces the far-IR emission in star-forming galaxies. The 350 mu m flux densities (15 < S(350) < 40 mJy) place these objects near the Herschel/SPIRE 350 mu m confusion threshold, with the lower limit on the star formation rate density suggesting the bulk of the 350 mu m contribution will come from less luminous infrared sources and normal galaxies. Therefore, the nature of the dominant source of the 350 mu m background-star-forming galaxies in the epoch of peak star formation in the universe-could be more effectively probed using ground-based instruments with their angular resolution and sensitivity offering significant advantages over space-based imaging.