Precise crop yield prediction provides valuable information for agricultural planning and decision-making processes. However, timely predicting crop yields remains challenging as crop growth is sensitive to growing season weather variation and climate change. In this work, we develop a deep learning-based solution, namely Multi-Modal Spatial-Temporal Vision Transformer (MMST-ViT), for predicting crop yields at the county level across the United States, by considering the effects of short-term meteorological variations during the growing season and the long-term climate change on crops. Specifically, our MMST-ViT consists of a Multi-Modal Transformer, a Spatial Transformer, and a Temporal Transformer. The Multi-Modal Transformer leverages both visual remote sensing data and short-term meteorological data for modeling the effect of growing season weather variations on crop growth. The Spatial Transformer learns the high-resolution spatial dependency among counties for accurate agricultural tracking. The Temporal Transformer captures the long-range temporal dependency for learning the impact of long-term climate change on crops. Meanwhile, we also devise a novel multi-modal contrastive learning technique to pre-train our model without extensive human supervision. Hence, our MMST-ViT captures the impacts of both short-term weather variations and long-term climate change on crops by leveraging both satellite images and meteorological data. We have conducted extensive experiments on over 200 counties in the United States, with the experimental results exhibiting that our MMST-ViT outperforms its counterparts under three performance metrics of interest. Our dataset and code are available at https://github.com/fudong03/MMST-ViT.