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We report on the first results of the POLAMI (Polarimetric Monitoring of AGNs with Millimetre Wavelengths) programme, a simultaneous 3.5 and 1.3mm full-Stokes-polarization monitoring of a sample of 36 of the brightest active galactic nuclei in the northern sky with the IRAM 30m telescope. Through a systematic statistical study of data taken from 2006 October (from 2009 December for the case of the 1.3mm observations) to 2014 August, we characterize the variability of the total flux density and linear polarization. We find that all sources in the sample are highly variable in total flux density at both 3.5 and 1.3 mm, as well as in spectral index, which (except in particularly prominent flares) is found to be optically thin between these two wavelengths. The total flux-density variability at 1.3mm is found, in general, to be faster, and to have larger fractional amplitude and flatter power-spectral-density slopes than at 3.5 mm. The polarization degree is on average larger at 1.3mm than at 3.5 mm, by a factor of 2.6. The variability of linear polarization degree is faster and has higher fractional amplitude than for total flux density, with the typical time-scales during prominent polarization peaks being significantly faster at 1.3mm than at 3.5 mm. The polarization angle at both 3.5 and 1.3mm is highly variable. Most of the sources show one or two excursions of >18° on time-scales from a few weeks to about a year during the course of our observations. The 3.5 and 1.3mm polarization angle evolution follows each other rather well, although the 1.3mm data show a clear preference to more prominent variability on the short time-scales, i.e. weeks. The data are compatible with multizone models of conical jets involving smaller emission regions for the shortest-wavelength emitting sites. Such smaller emitting regions should also be more efficient in energising particle populations, as implied by the coherent evolution of the spectral index and the total flux density during flaring activity of strong enough sources. The data also favour the integrated emission at 1.3mm to have better ordered magnetic fields than the one at 3.5 mm. [ABSTRACT FROM AUTHOR]