In the advent of next-generation wireless communication, millimeter-wave (mmWave) and terahertz (THz) technologies are pivotal for their high data rate capabilities. However, their reliance on large antenna arrays and narrow directive beams for ensuring adequate receive signal power introduces significant beam training overheads. This becomes particularly challenging in supporting highly-mobile applications such as drone communication, where the dynamic nature of drones demands frequent beam alignment to maintain connectivity. Addressing this critical bottleneck, our paper introduces a novel machine learning-based framework that leverages multi-modal sensory data, including visual and positional information, to expedite and refine mmWave/THz beam prediction. Unlike conventional approaches that solely depend on exhaustive beam training methods, our solution incorporates additional layers of contextual data to accurately predict beam directions, significantly mitigating the training overhead. Additionally, our framework is capable of predicting future beam alignments ahead of time. This feature enhances the system's responsiveness and reliability by addressing the challenges posed by the drones' mobility and the computational delays encountered in real-time processing. This capability for advanced beam tracking asserts a critical advancement in maintaining seamless connectivity for highly-mobile drones. We validate our approach through comprehensive evaluations on a unique, real-world mmWave drone communication dataset, which integrates concurrent camera visuals, practical GPS coordinates, and mmWave beam training data...