Basic Differential Geometry

You may need to consult background material from time to time. Here are a few basic texts that you might find helpful

M. D. Spivak, Calculus on Manifolds.
A concise introduction to the basics of coordinate-free calculus. Contains an introduction to manifolds, differential forms, and Stokes' Theorem.
M. D. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. I–V.
Not entirely comprehensive, of course, but does a good job of exposing basic manifold theory, Riemannian geometry, the calculus of variations, geometric PDE, and characteristic classes.
Manfredo do Carmo, Riemannian Geometry
A more focussed introduction to the subject.
John Milnor, Morse Theory.
Besides being a wonderful introduction to the subject, this is a masterpiece of mathematical exposition. Deserves a place on the bookshelf of every geometer and every topologist.
R. Bott and L. Tu, Differential Forms in Algebraic Topology.
Useful for gaining basic expertise in calculation with differential forms and their applications in computations in differential topology.
Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces.
Everything you want to know about the title subjects.

Symplectic Geometry

V. I. Arnold, Mathematical Methods in Classical Mechanics.
An excellent introduction to the classical point of view in Lagrangian and Hamiltonian mechanics.
R. L. Bryant An Introduction to Lie Groups and Symplectic Geometry.
Very basic and stripped down, so it will be used for the very beginnning of the subject. It is available in .pdf form here.
Y. Eliashberg and L. Traynor (eds.), Symplectic Geometry and Topology.
An excellent collection of introductory lectures to various areas in symplectic geometry and topology. These were delivered at the 1997 Park City/IAS Mathematics Institute.
M. Gromov, Pseudo-holomorphic curves on almost complex manifolds.
The classic 1985 Inventiones paper that launched the symplectic revolution. Although it is difficult to read for detail, its overview and motivating discussions are still an inspiration.
D. McDuff and D. Salamon, Introduction to Symplectic Topology.
The basic text for the majority of the present course.
A. Weinstein, Lectures on Symplectic Manifolds.
This excellent set of lectures from 1976 is a classic and still one of the best concise introductions to the subject.