We perform spatially resolved stellar population analysis for a sample of 69 early-type galaxies (ETGs) from the CALIFA integral field spectroscopic survey, including 48 ellipticals and 21 S0's. We generate and quantitatively characterize profiles of light-weighted mean stellar age and metallicity within ≲2Re, as a function of radius and stellar-mass surface density μ*. We study in detail the dependence of profiles on galaxies' global properties, including velocity dispersion σe, stellar mass, morphology. ETGs are universally characterized by strong, negative metallicity gradients (̃ -0.3 dex per Re) within 1Re, which flatten out moving towards larger radii. A quasi-universal local μ*-metallicity relation emerges, which displays a residual systematic dependence on σe, whereby higher σe implies higher metallicity at fixed μ*. Age profiles are typically U-shaped, with minimum around 0.4 Re, asymptotic increase to maximum ages beyond ̃ 1.5 Re, and an increase towards the centre. The depth of the minimum and the central increase anticorrelate with σe. A possible qualitative interpretation of these observations is a two-phase scenario. In the first phase, dissipative collapse occurs in the inner 1Re, establishing a negative metallicity gradient. The competition between the outside-in quenching due to feedback-driven winds and some form of inside-out quenching, possibly caused by central AGN feedback or dynamical heating, determines the U-shaped age profiles. In the second phase, the accretion of ex-situ stars from quenched and low-metallicity satellites shapes the flatter stellar population profiles in the outer regions.