Agile and reliable alignment of transceiver beams is crucial to support high transmission rate in millimeter-wave (mmWave) communications. In this letter, a deep learning aided two-stage multi-finger beam training (DL-TSMBT) algorithm is proposed for beam alignment purpose. In the first stage, a multi-finger beam based coarse scanning strategy is proposed to take a limited number of initial measurements, which are then fed into a customized convolutional neural network for feature extraction and candidate beam selection. In the second stage, the candidate beams are further trained to refine the beam selection. Numerical results validate the effectiveness of the DL-TSMBT proposed and show that DL-TSMBT outperforms several state-of-the-art traditional beam training baselines and a data-driven wide-beam training baseline, both in terms of misalignment probability and achievable spectrum efficiency.