We present high-quality, Keck spectroscopic data for a sample of 20 globular clusters (GCs) in the massive E0 galaxy NGC1407. A subset of twenty line-strength indices of the Lick/IDS system have been measured for both the GC system and the central integrated star-light of the galaxy. Ages, metallicities and [alpha/Fe] ratios have been derived using several different approaches. The majority GCs in NGC1407 studied are old, follow a tight metallicity sequence reaching values slightly above solar, and exhibit mean [alpha/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 some MR GCs exhibit significantly larger [Mg/Fe] and [C/Fe] ratios. Different star formation time-scales are proposed to explain the correlation between Mg and C abundances. We also find striking CN overabundances over the entire GC metallicity range. Interestingly, the behavior of C and N in metal-poor (MP) GCs clearly deviates from the one 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.