This graduate-level course equips mechanical and civil engineering students with the foundations and practical skills to harness Agentic AI for scientific research. Topics begin with Large Language Model (LLM) architecture and advanced prompt engineering, then progress to autonomous agent workflows using frameworks such as LangChain and AutoGen. A central focus is Scientific Machine Learning (SciML), including Physics-Informed Neural Networks (PINNs) and Neural Operators (PINOs) that serve as surrogate models for simulations in structural analysis, heat transfer, and fluid dynamics. The course culminates in multi-agent systems that autonomously propose, execute, and evaluate engineering simulations — mirroring the emerging paradigm of self-driving laboratories. Case studies are drawn from materials discovery, structural health monitoring, and computational fluid dynamics. Every lecture concept is reinforced through hands-on lab exercises; by course end, students will have built a portfolio of functional AI agent systems and a capstone project integrating an LLM with a domain-specific simulation tool.