Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.