We derive oxygen, sodium, iron, vanadium, and scandium abundances for giants in the intermediately metal-poor globular clusters M3 and M13 and for giants of comparable metallicity in the local halo field. The data are obtained from Lick Observatory Hamilton Echelle spectra centered on the [O I] doublet. For M13 we derive [[Fe/H]] = - 1.51 +/- 0.01 from 13 giants with a range of 500 K in effective temperature. For M3, we find [[Fe/H]] = - 1.47 +/- 0.01, from seven stars with a range of 300 K in effective temperature. There is no compelling evidence for star-to-star variations in [Fe/H] in either cluster. M13 stars divide into three apparently discrete oxygen abundance groups: four with "high" oxygen ([[O/Fe]] congruent-to + 0.3), seven with "low" oxygen ([[O/Fe]] congruent-to - 0.15), and two with "super-low" oxygen ([[O/Fe]] congruent-to - 0.8). Five of the seven M3 stars studied belong to the oxygen-rich group and two to the oxygen-poor group; no super-low oxygen stars have been detected in M3. Of the 16 intermediately metal-poor halo field giants analyzed here, 15 belong to the high-oxygen group. In M3 and M13, sodium abundances are found to be anticorrelated with oxygen abundances and in M13 they are found to be positively correlated with nitrogen abundances. Cluster and field stars appear to exhibit a "Na-O" signature (and a corresponding "Na-N" signature): M13 stars for the most part are O poor and Na rich, whereas M3 stars and halo field giants are 0 rich and Na poor. These results are compatible with the idea that there were "primordial" variations in the material out of which the present M13 and M3 low-mass stars were formed. The super-low oxygen stars may well have descended from stars like the low-oxygen stars studied here, as a result of C, O --> N nucleosynthesis and deep mixing. Thus it seems likely that a combination of primordial and mixing scenarios is required to explain the abundance distribution of CNO elements in intermediately metal-poor low-mass giants. Combining these results with those of our earlier study of M92, M15, and the very metal-poor halo field yields 59 giants with - 3 less-than-or-equal-to [Fe/H] less-than-or-equal-to - 1.3. The O-rich/O-poor dichotomy persists in M92, and most of the field halo giants (25 out of 27) belong to the O-rich group. [O/Fe] is essentially the same in O-rich giants of M3, M13, and the halo field, and a plot of [O/Fe] vs [Fe/H] is essentially flat for O-rich stars. Since the majority of M3 stars and field stars are O rich, the comparison of the main sequence of M3's C-M diagram with that of field subdwarfs having known trigonometric parallaxes should present no problem in principle, and thus the derivation of the age of M3 should be straightforward, in principle. We suggest that the similarity in the O-Na signature, CN signature, and HB morphology between stars of M3 and the halo field implies that M3 and the halo field stars have a similar age and chemical history. The case of M13, for which the majority of stars are O poor, is more problematical. Several lines of evidence from stellar evolution theory as well as the difference in O signature cited here, suggest that M13 is older than M3.