We analyse the relationships between galaxy morphology, colour, environment and stellar mass using data for over 100,000 objects from Galaxy Zoo, the largest sample of visually classified morphologies yet compiled. We conclusively show that colour and morphology fractions are very different functions of environment. Both are sensitive to stellar mass; however, at fixed stellar mass, while colour is also highly sensitive to environment, morphology displays much weaker environmental trends. Only a small part of both relations can be attributed to variation in the stellar mass function with environment. Galaxies with high stellar masses are mostly red, in all environments and irrespective of their morphology. Low stellar-mass galaxies are mostly blue in low-density environments, but mostly red in high-density environments, again irrespective of their morphology. The colour-density relation is primarily driven by variations in colour fractions at fixed morphology, in particular the fraction of spiral galaxies that have red colours, and especially at low stellar masses. We demonstrate that our red spirals primarily include galaxies with true spiral morphology. We clearly show there is an environmental dependence for colour beyond that for morphology. Before using the Galaxy Zoo morphologies to produce the above results, we first quantify a luminosity-, size- and redshift-dependent classification bias that affects this dataset, and probably most other studies of galaxy population morphology. A correction for this bias is derived and applied to produce a sample of galaxies with reliable morphological type likelihoods, on which we base our analysis.