We present high-quality Keck spectroscopic data for a sample of 20 globular clusters (GCs) in the massive E0 galaxy NGC 1407. A subset of 20 line-strength indices of the Lick/IDS system has been measured for both the GC system and the central integrated starlight of the galaxy. Ages, metallicities, and [α/Fe] ratios have been derived using several different approaches. The majority of GCs in NGC 1407 studied are old, follow a tight metallicity sequence reaching values slightly above solar, and exhibit mean [α/Fe] ratios of ~0.3 dex. In addition, three GCs are formally derived to be young (~4 Gyr), but we argue that they are actually old GCs hosting blue horizontal branches. We report, for the first time, evidence for the existence of two chemically distinct subpopulations of metal-rich (MR) GCs. We find that some MR GCs exhibit significantly larger [Mg/Fe] and [C/Fe] ratios. Different star formation timescales are proposed to explain the correlation between Mg and C abundances. We also find striking CN overabundances over the entire GC metallicity range. It is interesting to note that the behavior of C and N in metal-poor GCs clearly deviates from that in MR GCs. In particular, for MR GCs, N increases dramatically while C essentially saturates. This may be interpreted as a consequence of the increasing importance of the CNO cycle with increasing metallicity.