I would like to know: what are some of the best introductory books to Lagrangian Mechanics?
Rather than a book, consider the Susskind Physics lectures made available on Youtube as well as Stanford on iTunes via iTunes-University. You want to watch the latest version of the course lectures on Classical Mechanics. After a small bit of introductory material, Susskind gets right into the Lagrangian and Hamiltonian approach to mechanics using a simple approach for easy understanding. Better yet, Susskind also covers topics like Liouville's Theorem, Poisson Brackets, and examples of applying Lagrangian methods to Maxwell's equations for Electromagnetic Field.
I almost forgot. Susskind's lectures are available on-line via the website located at: http://theoreticalminimum.com/home. Take that link, jump to the menu item Courses, and begin watching the lectures on Classical Mechanics. Each lecture is roughly 2 hours long, entertaining, and at a reasonably descent level consistent with advanced undergraduate courses in Classical Mechanics.
Susskind also has a book that parallels this particular video series called "The Theoretical Minimum". I believe that this name, "The Theoretical Minimum" comes from the Russian physicist L.D. Landau who preferred to welcome students into his graduate programs who had already demonstrated knowledge of the theoretical minimum amount of physics to succeed in his graduate PhD programs.
By the way, the book Mechanics, by L.D. Landau and E.M. Lifshitz is excellent too but more towards a graduate level of mechanics. It starts out with Lagrangian methods in chapter 1 (generalized coordinates, principle of least action, Lagrangian, Euler-Lagrange equation, etc.).
There are many other book recommendations that can be found here on Physics S.E. but I do believe that the best starting place is Susskind's lectures.