With the increase in the number of optimization objectives, balancing the convergence and diversity in evolutionary multiobjective optimization becomes more intractable. So far, a variety of evolutionary algorithms have been proposed to solve many-objective optimization problems (MaOPs) with more than three objectives. Most of the existing algorithms, however, find difficulties in simultaneously counterpoising convergence and diversity during the whole evolutionary process. To address the issue, this paper proposes to solve MaOPs via multistage evolutionary search. To be specific, a two-stage evolutionary algorithm is developed, where the convergence and diversity are highlighted during different search stages to avoid the interferences between them. The first stage pushes multiple subpopulations with different weight vectors to converge to different areas of the Pareto front. After that, the nondominated solutions coming from each subpopulation are selected for generating a new population for the second stage. Moreover, a new environmental selection strategy is designed for the second stage to balance the convergence and diversity close to the Pareto front. This selection strategy evenly divides each objective dimension into a number of intervals, and then one solution having the best convergence in each interval will be retained. To assess the performance of the proposed algorithm, 48 benchmark functions with 7, 10, and 15 objectives are used to make comparisons with five representative many-objective optimization algorithms.