We present an adaptive goal-oriented approach to projection-based model reduction of parametrized partial differential equations (PDEs), with applications to aerodynamic flows with shocks. We emphasize efficient approximation, reliable error estimation, and systematic error control for both full-order model (FOM) and reduced-order model (ROM). The key ingredients are the following: (i) a FOM based on high-order discontinuous Galerkin method with PDE-smoothed artificial viscosity model for robust shock capturing; (ii) a ROM that builds on registration-based nonlinear approximation spaces for efficient reduction of parameter-dependent discontinuities and hyperreduction based on empirical quadrature procedure, with dual-based output correction for superconvergent prediction; (iii) dual-weighted residual (DWR) error estimation and anisotropic mesh refinement; and (iv) a higher-order variant of DWR error estimate for the superconvergent ROM and the associated greedy sampling algorithm. We place a particular emphasis on understanding and effectively treating the interactions between the modified PDEs with artificial viscosity stabilization, DWR error estimate, and dual-based superconvergent output correction. We demonstrate the formulation using parametrized transonic aerodynamics problems governed by the compressible Euler and Reynolds-averaged Navier--Stokes equations.