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 S0s. We generate and quantitatively characterize profiles of light-weighted mean stellar age and metallicity within $lesssim 2R_e$, as a function of radius and stellar-mass surface density $mu_*$. We study in detail the dependence of profiles on galaxies global properties, including velocity dispersion $sigma_e$, stellar mass, morphology. ETGs are universally characterized by strong, negative metallicity gradients ($sim -0.3,text{dex}$ per $R_e$) within $1,R_e$, which flatten out moving towards larger radii. A quasi-universal local $mu_*$-metallicity relation emerges, which displays a residual systematic dependence on $sigma_e$, whereby higher $sigma_e$ implies higher metallicity at fixed $mu_*$. Age profiles are typically U-shaped, with minimum around $0.4,R_e$, asymptotic increase to maximum ages beyond $sim 1.5,R_e$, and an increase towards the centre. The depth of the minimum and the central increase anti-correlate with $sigma_e$. A possible qualitative interpretation of these observations is a two-phase scenario. In the first phase, dissipative collapse occurs in the inner $1,R_e$, 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.
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