We present spectral observations of 130 young stellar objects (YSOs) in the Serpens Cloud Core and NGC 1333 embedded clusters. The observations consist of near-IR spectra in the H and K bands from SpeX on the IRTF and far-red spectra (6000-9000 Å) from Hectospec on the Multi-Mirror Telescope. These YSOs were identified in previous Spitzer and Chandra observations, and the evolutionary classes of the YSOs were determined from the Spitzer mid-IR photometry. With these spectra we search for corroborating evidence for the pre-main-sequence nature of the objects, study the properties of the detected emission lines as a function of evolutionary class, and obtain spectral types for the observed YSOs. The temperatures implied by the spectral types are combined with luminosities determined from the near-IR photometry to construct Hertzsprung-Russell (H-R) diagrams for the clusters. By comparing the positions of the YSOs in the H-R diagrams with the pre-main-sequence tracks of Baraffe (1998), we determine the ages of the embedded sources and study the relative ages of the YSOs with and without optically thick circumstellar disks. The apparent isochronal ages of the YSOs in both clusters range from less than 1 Myr to 10 Myr, with most objects below 3 Myr. The observed distributions of ages for the Class II and Class III objects are statistically indistinguishable. We examine the spatial distribution and extinction of the YSOs as a function of their isochronal ages. We find the sources <3 Myr to be concentrated in the molecular cloud gas, while the older sources are spatially dispersed and are not deeply embedded. Nonetheless, the sources with isochronal ages >3 Myr show all the characteristics of YSOs in their spectra, their IR spectral energy distributions, and their X-ray emission; we find no evidence that they are contaminating background giants or foreground dwarfs. However, we find no corresponding decrease in the fraction of sources with infrared excess with isochronal age; this suggests that the older isochronal ages may not measure the true age of the >3 Myr YSOs. Thus, the nature of the apparently older sources and their implications for cluster formation remain unresolved.