Recent advancements in multimodal Human-Robot Interaction (HRI) datasets have highlighted the fusion of speech and gesture, expanding robots' capabilities to absorb explicit and implicit HRI insights. However, existing speech-gesture HRI datasets often focus on elementary tasks, like object pointing and pushing, revealing limitations in scaling to intricate domains and prioritizing human command data over robot behavior records. To bridge these gaps, we introduce NatSGD, a multimodal HRI dataset encompassing human commands through speech and gestures that are natural, synchronized with robot behavior demonstrations. NatSGD serves as a foundational resource at the intersection of machine learning and HRI research, and we demonstrate its effectiveness in training robots to understand tasks through multimodal human commands, emphasizing the significance of jointly considering speech and gestures. We have released our dataset, simulator, and code to facilitate future research in human-robot interaction system learning; access these resources at https://www.snehesh.com/natsgd/