Machine learning methods have been widely used in different applications, including process control and monitoring. For handling statistical process control (SPC) problems, conventional supervised machine learning methods (e.g., artificial neural networks and support vector machines) would have some difficulties. For instance, a training dataset containing both in-control and out-of-control process observations is required by a supervised machine learning method, but it is rarely available in SPC applications. Furthermore, many machine learning methods work like black boxes. It is often difficult to interpret their learning mechanisms and the resulting decision rules in the context of an application. In the SPC literature, there have been some existing discussions on how to handle the lack of out-of-control observations in the training data, using the one-class classification, artificial contrast, real-time contrast, and some other novel ideas. However, these approaches have their limitations in handling SPC problems. In this paper, we extend the self-starting process monitoring idea that has been employed widely in modern SPC research to a general learning framework for monitoring processes with serially correlated data. Under the new framework, process characteristics to learn are well specified in advance, and process learning is sequential in the sense that the learned process characteristics keep being updated during process monitoring. The learned process characteristics are then incorporated into a control chart for detecting process distributional shifts based on all available data by the current observation time. Numerical studies show that process monitoring based on the new learning framework is more reliable and effective than some representative existing machine learning SPC approaches.