We are concerned with stochastic partial differential equations appearing in the modeling of non-Newtonian fluids. More precisely, we study models such as p-Navier--Stokes system where randomness is encoded in two ways: in the form of a random initial datum and in the form of stochastic external forces given by a general (nonlinear) multiplicative noise. Within the first part, we aim at developing a regularity theory. In particular, we are interested in the natural regularity which is essential for numerical approximations. The second part of our project is concerned with developing efficient numerical schemes and proving their rates of convergence.