We present measurements of the Nii/Halpha ratio as a probe of gas-phase oxygen abundance for a sample of 419 star-forming galaxies at z= 0.6-2.7 from the KMOS super(3D) near-IR multi-integral field unit (IFU) survey. The mass-metallicity relation (MZR) is determined consistently with the same sample selection, metallicity tracer, and methodology over the wide redshift range probed by the survey. We find good agreement with long-slit surveys in the literature, except for the low-mass slope of the relation at z~ 2.3, where this sample is less biased than previous samples based on optical spectroscopic redshifts. In this regime we measure a steeper slope than some literature results. Excluding the contribution from active galactic nuclei from the MZR reduces sensitivity at the high-mass end, but produces otherwise consistent results. There is no significant dependence of the Nii/Halpha ratio on star formation rate at fixed redshift and stellar mass. The IFU data allow spatially resolved measurements of Nii/Halpha, from which we can infer abundance gradients for 180 galaxies, thus tripling the current sample in the literature. The observed gradients are on average flat, with only 15 gradients statistically offset from zero at > 3sigma. We have modeled the effect of beam smearing, assuming a smooth intrinsic radial gradient and known seeing, inclination, and effective radius for each galaxy. Our seeing-limited observations can recover up to 70% of the intrinsic gradient for the largest, face-on disks, but only 30% for the smaller, more inclined galaxies. We do not find significant trends between observed or corrected gradients and any stellar population, dynamical, or structural galaxy parameters, mostly in agreement with existing studies with much smaller sample sizes. In cosmological simulations, strong feedback is generally required to produce flat gradients at high redshift.