In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of $p-$Laplacian type. The problem in its variational form is as follows: $$ \displaystyle \text{min} \left{ \int\limits\_{\Omega \cap {v>0}} \left(\frac{1}{p}|\nabla v|^p + \lambda\_{+}^p+ f\_{+}v \right)dx + \int\limits\_{\Omega \cap {v\leq 0}} \left(\frac{1}{q}|\nabla v|^q + \lambda\_{-}^q+ f\_{-}v\right)dx \right}. $$ Here we minimize among all admissible functions $v$ in an appropriate Sobolev space with a prescribed boundary datum $v=g$ on $\partial \Omega$. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where $p$ and $q$ go to infinity, obtaining a limiting free boundary problem governed by the $\infty$-Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.