We investigate the temporal and colour variability of 897 blazars, comprising 455 BL Lacertae objects (BL Lacs) and 442 Flat Spectrum Radio Quasars (FSRQs), selected from the Roma-BZCAT catalogue, using the multiband light curves from the Zwicky Transient Facility (ZTF DR6) survey. Assessing the colour variability characteristics over ~2 yr time-scales, we found that 18.5 per cent (84 out of 455) BL Lacs showed a stronger bluer-when-brighter (BWB) trend, whereas 9.0 per cent (41 out of 455) showed a redder-when-brighter (RWB) trend. The majority (70 per cent) of the BL Lacs showing RWB are host galaxy dominated. For the FSRQ subclass, 10.2 per cent (45 out of 442) objects showed a strong BWB trend and 17.6 per cent (78 out of 442) showed a strong RWB trend. Hence, we find that BL Lacs more commonly follow a BWB trend than do FSRQs. This can be attributed to the more dominant jet emission in the case of BL Lacs and the contribution of thermal emission from the accretion disc for FSRQs. In analysing the colour behaviour on shorter time windows, we find many blazars evince shorter partial trends of BWB or RWB nature (or occasionally both). Some of such complex colour behaviours observed in the colour-magnitude diagrams of the blazars may result from transitions between the jet-dominated state and the disc-dominated state and vice versa.