Can someone please tell me some introductory book on symplectic geometry? I have no prior idea of the subject but I do know about Lagrangian and Hamiltonian dynamics (at the level of Landau-Lifshitz Vol. 1). Thanks in advance. :-)
If you are physically inclined, V.I.Arnold's Mathematical methods of classical mechanics provides a masterful short introduction to symplectic geometry, followed by a wealth of its applications to classical mechanics. The exposition is much more systematic than vol 1 of Landau and Lifschitz and, while mathematically sophisticated, it is also very lucid, demonstrating the interaction between physical ideas and mathematical concepts that support them. (It is also worth mentioning that Arnold was largely responsible for the reawakening of interest to symplectic geometry at the end of 1960s and pioneered the study of symplectic topology. Some of these developments were brand new when the book was first published in 1974 and are briefly discussed in the appendices).
In addition to the notes by Cannas da Silva mentioned by Dick Palais, here are further two advanced books covering somewhat different territory:
Michèle Audin, Torus actions on symplectic manifolds (2nd edition)A
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology
A In her book, Michèle Audin herself recommends
Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics
as a wonderful introduction to symplectic geometry.
My favourite book on symplectic geometry is "Symplectic Invariants and Hamiltonian Dynamics" by Hofer and Zehnder. It's wonderfully written. Another lovely book (which has just been reissued as an AMS Chelsea text) is Abraham and Marsden's book "Foundations of Mechanics" which covers a lot of symplectic geometry as well as so much more...
Sternberg and Guillemin's Symplectic Techniques in Physics is one of a kind. In spite of the name it feels more like a text on mathematics than on physics, with the exception of the first motivating section of the book.
Arnold's book that Victor recommends is also one of my favorites. But much of it covers the kind of material you might find in Goldstein or Landau-Lifschitz, albeit treated from a more sophisticated and geometric point of view. If you already have that thoroughly mastered, Sternberg and Guillemin might be more what you want, especially the later parts.
For a more Lie-group focused account, you can try Robert Bryant's lectures on Lie groups and symplectic geometry which are available online here. In the final lecture he describes the h-principle and others ideas of Gromov in symplectic geometry, like pseudo-holomorphic curves.
A classic, physically inclined introduction is http://www.springer.com/us/book/9780817636951 (Structure of Dynamical Systems: A Symplectic View of Physics, by J.M. Souriau).
You can also try the book An introduction to symplectic geometry by Rolf Berndt which should be a good fit given your prerequisites.
Yet another possibility could be A Brief Introduction to Symplectic and Contact Manifolds by Banyaga and Houenou. Also Hamiltonian dynamics by Vilasi could be helpful.