Social media platforms generate massive volumes of heterogeneous data, capturing user behaviors, textual content, temporal dynamics, and network structures. Analyzing such data is crucial for understanding phenomena such as opinion dynamics, community formation, and information diffusion. However, discovering insights from this complex landscape is exploratory, conceptually challenging, and requires expertise in social media mining and visualization. Existing automated approaches, though increasingly leveraging large language models (LLMs), remain largely confined to structured tabular data and cannot adequately address the heterogeneity of social media analysis. We present SIA (Social Insight Agents), an LLM agent system that links heterogeneous multi-modal data -- including raw inputs (e.g., text, network, and behavioral data), intermediate outputs, mined analytical results, and visualization artifacts -- through coordinated agent flows. Guided by a bottom-up taxonomy that connects insight types with suitable mining and visualization techniques, SIA enables agents to plan and execute coherent analysis strategies. To ensure multi-modal integration, it incorporates a data coordinator that unifies tabular, textual, and network data into a consistent flow. Its interactive interface provides a transparent workflow where users can trace, validate, and refine the agent's reasoning, supporting both adaptability and trustworthiness. Through expert-centered case studies and quantitative evaluation, we show that SIA effectively discovers diverse and meaningful insights from social media while supporting human-agent collaboration in complex analytical tasks.