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 Lieberman Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics
as a wonderful introduction to symplectic geometry.