The quasar 3C 279 was the target of an extensive multiwavelength monitoring campaign from 2006 January through April. An optical-IR-radio monitoring campaign by the Whole Earth Blazar Telescope (WEBT) collaboration was organized around target-of-opportunity X-ray and soft γ-ray observations with Chandra and INTEGRAL in 2006 mid-January, with additional X-ray coverage by RXTE and Swift XRT. In this paper we focus on the results of the WEBT campaign. The source exhibited substantial variability of optical flux and spectral shape, with a characteristic timescale of a few days. The variability patterns throughout the optical BVRI bands were very closely correlated with each other, while there was no obvious correlation between the optical and radio variability. After the ToO trigger, the optical flux underwent a remarkably clean quasi-exponential decay by about 1 mag, with a decay timescale of τd ~ 12.8 days. In intriguing contrast to other (in particular, BL Lac type) blazars, we find a lag of shorter wavelength behind longer wavelength variability throughout the RVB wavelength ranges, with a time delay increasing with increasing frequency. Spectral hardening during flares appears delayed with respect to a rising optical flux. This, in combination with the very steep IR-optical continuum spectral index of α0 ~ 1.5-2.0, may indicate a highly oblique magnetic field configuration near the base of the jet, leading to inefficient particle acceleration and a very steep electron injection spectrum. An alternative explanation through a slow (timescale of several days) acceleration mechanism would require an unusually low magnetic field of B 0.2 G, about an order of magnitude lower than inferred from previous analyses of simultaneous SEDs of 3C 279 and other flat-spectrum radio quasars with similar properties.