In this article, we continue the investigations initiated by the first author in Balan (2015) related to the study of stochastic partial differential equations (SPDEs) with Lévy colored noise on $\mathbb{R}_{+} \times \mathbb{R}^d$. This noise is constructed from a Lévy white noise (which is in turn built from a Poisson random measure with intensity $dtdx ν(dz)$), using the convolution with a suitable spatial kernel $κ$. We assume that the Lévy measure $ν$ has finite variance. Therefore, the stochastic integral with respect to this noise is constructed similarly to the integral with respect to the spatially-homogeneous Gaussian case considered in Dalang (1999). Using Rosenthal's inequality, we provide an upper bound for the $p$-th moment of the stochastic integral with respect to the Lévy colored noise, which allows us to identify sufficient conditions for the solution of an SPDE driven by this noise to have higher order moments. We first analyze this question for the linear SPDE, considering as examples the stochastic heat and wave equations in any dimension $d$, for three examples of kernels $κ$: the heat kernel, the Riesz kernel, and the Bessel kernel. Then, we present a general theory for a non-linear SPDE with Lipschitz coefficients, and perform a detailed analysis in the case of the heat equation (in dimension $d\geq 1$), and wave equation (in dimension $d\leq 3$), for the same kernels $κ$. We show that the solution of each of these equations has a finite upper Lyapounov exponent of order $p\geq 2$, and in some cases, is weakly intermittent (in the sense of Foondun and Khoshnevisan, 2013). In the case of the parabolic/hyperbolic Anderson model with Lévy colored noise, we provide the Poisson chaos expansion of the solution and the explicit form of the second-order Lyapounov exponent.