This article concerns the dynamic transitions of a non-Newtonian horizontal fluid layer with thermal and solute diffusion and in the presence of vertical magnetic field. First, a linear stability analysis is done by deriving the principle of exchange of stability condition, which shows the system loses stability when thermal Rayleigh number exceeds a critical threshold. Second, we considered the transition induced by real eigenvalues and complex eigenvalues, respectively, and two nonlinear transition theorems along with several transition numbers determining the transition types are obtained via the method of center manifold reduction. Finally, rigorous numerical computations are performed to offer examples of possible transitions, as well as the stable convection patterns. Our results show that when the diffusivities from big to small are thermal, solute concentration and magnetic diffusion, both continuous and jump transitions can occur for certain parameters; and if the diffusivities from big to small is the inverse of the previous case, only continuous transition induced by real eigenvalues are observed, which indicate a stationary convection.