Ocean fronts are a response to the variabilities of marine hydrographic elements and are an important mesoscale ocean phenomenon, playing a significant role in fish farming and fishing, sea-air exchange, marine environmental protection, etc. The horizontal gradients of sea surface temperature (SST) are frequently applied to reveal ocean fronts. Up to now, existing spatiotemporal prediction approaches have suffered from low prediction precision and poor prediction quality for non-stationary data, particularly for long-term prediction. It is a challenging task for medium- and long-term fine-grained prediction for SST sequences and fronts in oceanographic research. In this study, SST sequences and fronts are predicted for future variation trends based on continuous mean daily remote sensing satellite of SST data. To enhance the precision of the predicted SST sequences and fronts, this paper proposes a novel memory-contextual spatiotemporal transfer network (MCSTNet) for SST sequence and front predictions. MCSTNet involves three components: the encoder-decoder structure, a time transfer module, and a memory-contextual module. The encoder-decoder structure is used to extract the rich contextual and semantic information in SST sequences and frontal structures from the SST data. The time transfer module is applied to transfer temporal information and fuse low-level, fine-grained temporal information with high-level semantic information to improve medium- and long-term prediction precision. And the memory-contextual module is employed to fuse low-level, spatiotemporal information with high-level semantic information to enhance short-term prediction precision. In the training process, mean squared error (MSE) loss and contextual loss are combined to jointly guide the training of MCSTNet. Extensive experiments demonstrate that MCSTNet predicts more authentic and reasonable SST sequences and fronts than the state-of-the-art (SOTA) models on the SST data.