We present the first step in the development of an adaptive finite element solver for the shallow water equations for coastal engineering applications, which also inherits the efficiency and robustness properties of the semi-implicit methods developed in in “Casulli, A high-resolution wetting and drying algorithm for free-surface hydrodynamics (2009)”. In this work we will restrict our attention to a second order IMEX time discretization that treats the pressure gradient term explicitly, while applying an implicit method to the friction term. The IMEX scheme is based on the combination of the stiffly accurate TR-BDF2 for the implicit part and an explicit three stages second order Runge-Kutta scheme specifically designed to match the coupling conditions. Within our finite element framework we focus on a robust and accurate treatment of the bathymetry, nowadays available with higher resolution than the mesh in coastal areas. This poses a series of challenges for higher order methods that work on quite coarse meshes: the mesh may not be aligned to large bathymetric gradients or jumps and the finite element method should be able to handle large gradients within an element or along an edge. We choose as prognostic variable the free-surface elevation which is smooth and for which we employ a finite element representation. The bathymetry at the quadrature node is directed evaluated from the reference data without any local modification. We discuss some numerical implications of this approach, for example concerning mass-conservation and the discretization of a passive tracer. We show the robustness in presence of a realistic bathymetry while, for the opposite choice of the water depth as prognostic variable, a smoothing or a TVD limiting of the bathymetry is necessary to not incur in the Gibbs phenomenon. The spatial discretization is based on a high order Discontinuous Galerkin (DG) method as implemented in the deal.II library. In this framework, we test non-conforming meshes with static and dynamic Adaptive Mesh Refinement (AMR) handled by deal.II, to simulate the tidal circulations in a complex coastal environment such as the Venice Lagoon. We show that AMR allows to resolve small-scale structures that are absent in the static runs. Finally we demonstrate the good parallel performance of the resulting discretization.