Memristor devices are considered to have the potential to implement unsupervised learning, especially spike timing-dependent plasticity (STDP), in the field of neuromorphic hardware research. In this study, a neuromorphic hardware system for multilayer unsupervised learning was designed, and unsupervised learning was performed with a memristor neural network. We showed that the nonlinear characteristic memristor neural network can be trained by unsupervised learning only with the correlation between inputs and outputs. Moreover, a method to train nonlinear memristor devices in a supervised manner, named guide training, was devised. Memristor devices have a nonlinear characteristic, which makes implementing machine learning algorithms, such as backpropagation, difficult. The guide-training algorithm devised in this paper updates the synaptic weights by only using the correlations between inputs and outputs, and therefore, neither complex mathematical formulas nor computations are required during the training. Thus, it is considered appropriate to train a nonlinear memristor neural network. All training and inference simulations were performed using the designed neuromorphic hardware system. With the system and memristor neural network, the image classification was successfully done using both the Hebbian unsupervised training and guide supervised training methods.