Traditional time series (TS) forecasting models are based on fixed, static datasets and lack scalability when faced with the continuous influx of data in real-world scenarios. Real-time online learning of data streams is crucial for improving forecasting efficiency. However, few studies focus on online TS forecasting, and existing approaches have several limitations. Most current online TS forecasting models merely train on data streams and are ineffective in handling concept drift scenarios. Furthermore, they often fail to adequately consider dependencies between variables and do not leverage the robust modeling capabilities of offline models. Therefore, we propose an innovative online learning method called OnsitNet. It consists of multiple learning modules that progressively expand the receptive field of convolutional kernels within the learning modules using an exponentially growing dilation factor, aiding in the capture of multi-scale data features. Within the learning modules, we propose an online learning strategy focusing on memorizing concept drift scenarios, with a fast learner, memorizer, and Pearson trigger. The Pearson trigger activates dynamic interaction between the fast learner and memorizer by detecting new data patterns, facilitating online rapid learning of data streams. To capture the dependencies between variables, we propose a new model, SITransformer, which is a streamlined version of the offline model ITransformer. Unlike the traditional Transformer, it reverses the roles of the feed-forward network and the attention mechanism. This inverted architecture is more effective at learning the correlations between variables. Experimental results on five real-world datasets show OnsitNet achieves lower online prediction errors, enabling timely and effective forecasting of future trends in TS data.