Seismic records, known as seismograms, are crucial records of ground motion resulting from seismic events, constituting the backbone of earthquake research and monitoring. The latest advancements in deep learning have significantly facilitated various seismic signal processing tasks. This paper introduces a novel backbone neural network model designed for various seismic monitoring tasks, named Seismogram Transformer (SeisT). Thanks to its efficient network architecture, SeisT matches or even outperforms the state-of-the-art models in earthquake detection, seismic phase picking, first-motion polarity classification, magnitude estimation, and azimuth estimation tasks, particularly in terms of out-of-distribution generalization performance. SeisT consists of multiple network layers composed of different foundational blocks, which help the model understand multi-level feature representations of seismograms from low-level to high-level complex features, effectively extracting features such as frequency, phase, and time-frequency relationships from input seismograms. Three different-sized models were customized based on these diverse foundational modules. Through extensive experiments and performance evaluations, this study showcases the capabilities and potential of SeisT in advancing seismic signal processing and earthquake research.