Nobody mentioned "Gilmore: Lie Groups, Physics, and Geometry" yet.

A very down to earth introduction with many examples and clear explanations. Especially targeted at physicists, engineers and chemists.

If you follow the above link you can read some sample chapters.

The cover summarizes the set up of the book quite neatly:

"Describing many of the most important aspects of Lie group theory, this book presents the subject in a ‘hands on’ way. Rather than concentrating on theorems and proofs, the book shows the relation of Lie groups with many branches of mathematics and physics, and illustrates these with concrete computations. Many examples of Lie groups and Lie algebras are given throughout the text, with applications of the material to physical sciences and applied mathematics. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

Robert Gilmore is a Professor in the Department of Physics at Drexel University, Philadelphia. He is a Fellow of the American Physical Society, and a Member of the Standing Committee for the International Colloquium on Group Theoretical Methods in Physics. His research areas include group theory, catastrophe theory, atomic and nuclear physics, singularity theory, and chaos."