Heterogeneous Information Networks (HINs) are labeled graphs that depict relationships among different types of entities (e.g., users, movies and directors). For HINs, meta-path-based recommenders (MPRs) utilize meta-paths (i.e., abstract paths consisting of node and link types) to predict user preference, and have attracted a lot of attention due to their explainability and performance. We observe that the performance of MPRs is highly sensitive to the meta-paths they use, but existing works manually select the meta-paths from many possible ones. Thus, to discover effective meta-paths automatically, we propose the Reinforcement learning-based Meta-path Selection (RMS) framework. Specifically, we define a vector encoding for meta-paths and design a policy network to extend meta-paths. The policy network is trained based on the results of downstream recommendation tasks and an early stopping approximation strategy is proposed to speed up training. RMS is a general model, and it can work with all existing MPRs. We also propose a new MPR called RMS-HRec, which uses an attention mechanism to aggregate information from the meta-paths. We conduct extensive experiments on real datasets. Compared with the manually selected meta-paths, the meta-paths identified by RMS consistently improve recommendation quality. Moreover, RMS-HRec outperforms state-of-the-art recommender systems by an average of 7% in hit ratio. The codes and datasets are available on https://github.com/Stevenn9981/RMS-HRec.