A semi-implicit and semi-Lagrangian Discontinuous Galerkin method for the shallow water equations is proposed and analyzed, based on semi-implicit and semi-Lagrangian techniques previously introduced separately by the authors, see Restelli, Bonaventura and Sacco, JCP, 2006 and Restelli and Giraldo, SIAM J. Sci. Comp., 2009. The method is equipped with a simple p-adaptivity criterion, that allows to adjust effectively the number of local degrees of freedom employed to the local structure of the solution. The wave dispersion and von Neumann stability analysis for the method are presented. Numerical results in the framework of one dimensional test cases prove that the method captures accurately and effectively the main features of linear gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear open channel flow tests. The effectiveness of the method is also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree. Perspectives for extensions to geophysical flows applications are discussed.