The variable-coefficient equations play an important role in mathematical physics and physical applications. However, it is very difficult to study exact solutions and other properties of such equations. In this paper, we investigate the complete integrability classification of a class of the generalized variable-coefficient nonlinear wave equation by the Painlevé method, the integrability and integrable conditions of the variable-coefficient equations are obtained, then the exact solutions are provided by the truncated expansion method. Moreover, some new types of explicit solutions to the nonlinear variable-coefficient equations are investigated by the invariant subspace method (ISM).