Cell-free millimeter-wave (mmWave) multiple-input multiple-output (MIMO) communications have been proposed as promising enablers for the next generation wireless networks. In this paper, we study the user association and hybrid beamforming in cell-free mmWave systems without full channel state information (CSI) acquisition. We consider a cloud radio access network (C-RAN), where multiple remote radio heads (RRHs) are distributed to communicate with users via analog beamforming, and connected to a centralized baseband unit (BBU) through fronthaul links which executes digital beamforming. We aim to jointly design user association, hybrid beamforming, and fronthaul compression with the aid of uplink training. A train-and-design framework is developed to achieve this goal. In particular, we first propose a two-stage uplink training approach to assist RRH-level design, during which the analog beamforming and user association are obtained. After that, digital beamforming and fronthaul compression are optimized at BBU based on the training results. Two performance metrics are considered in this paper, i.e. weighted sum-rate maximization and max-min fairness. Simulation results demonstrate the effectiveness of the proposed train-and-design framework for both sum-rate maximization and max-min fairness performance metrics. It is shown that the proposed algorithms can achieve comparable performance to the full-digital beamformer.