We study the Signorini problem in the fully mixed form with stress as a new unknown. We reformulate the continuous and discrete mixed variational inequalities into the projection problems and show the well-posedness. Two Lagrange multiplier formulations are presented in continuous and discrete senses. The error estimates for the discrete mixed variational inequality are developed. To solve the fully mixed variational inequality, we design two Active/Inactive set algorithms. Moreover, we study the penalty approaches in the fully mixed form. Two mixed penalty formulations are introduced. We propose two iteration schemes to solve penalty problems and analyze their convergence. The well-posedness and error analysis for the mixed penalty problems is established for both continuous and discrete scenarios. We verify the theoretical convergence rates of the finite element discretization and the iteration algorithms with numerical experiments.