In the intracluster medium (ICM) of galaxy clusters, heat and momentum are transported almost entirely along (but not across) magnetic field lines. We perform the first fully self-consistent Braginskii magnetohydrodynamics (MHD) simulations of galaxy clusters including both of these effects. Specifically, we perform local and global simulations of the magnetothermal instability (MTI) and the heat-flux-driven buoyancy instability (HBI) and assess the effects of viscosity on their saturation and astrophysical implications. We find that viscosity has only a modest effect on the saturation of the MTI. As in previous calculations, we find that the MTI can generate nearly sonic turbulent velocities in the outer parts of galaxy clusters, although viscosity somewhat suppresses the magnetic field amplification. At smaller radii in cool-core clusters, viscosity can decrease the linear growth rates of the HBI. However, it has less of an effect on the HBI’s non-linear saturation, in part because three-dimensional interchange motions (magnetic flux tubes slipping past each other) are not damped by anisotropic viscosity. In global simulations of cool-core clusters, we show that the HBI robustly inhibits radial thermal conduction and thus precipitates a cooling catastrophe. The effects of viscosity are, however, more important for higher entropy clusters. We argue that viscosity can contribute to the global transition of cluster cores from cool-core to non-cool-core states: additional sources of intracluster turbulence, such as can be produced by active galactic nuclei feedback or galactic wakes, suppress the HBI, heating the cluster core by thermal conduction; this makes the ICM more viscous, which slows the growth of the HBI, allowing further conductive heating of the cluster core and a transition to a non-cool-core state.