Book about classical mechanics

Maybe no symplectic geometry or forms here, but this book has a LOT to offer:

http://www.people.fas.harvard.edu/~djmorin/book.html

Pure beaten gold, any edition, paperback. It never leaves you. Lagrangian approach.
See the reviews on Amazon L D Landau (Author), E.M. Lifshitz (Author) http://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960

My favourite for pure classical mechanics is generally the book by Goldstein which includes the Lagrangian and Hamiltonian methods (although I'm not sure about symplectic geometrical and mathematical foundations).

If you want a firm look at curved spacetime manifolds (for vectors, one forms, tangent bundles etc.) I would recommend Carroll's Spacetime and Geometry but it deals with the mathematical backing for General Relativity, which is classical mechanics, but on a curved spacetime.

I can't believe nobody's mentioned Arnol'd's book "Mathematical Methods for Classical Mechanics" - it covers everything you ask for in the first paragraph quite elegantly (though sometimes somewhat tersely).

As already mentioned the standard introductory books in hamiltonian geometrical (point-) mechanics are Foundations of Mechanics by Abraham and Marsden and Arnolds Mathematical Methods of Classical Mechanics. Another standard book is: "Classical Mathematical Physics" by Walter Thirring.

You may also have a look at "Symmetry in mechanics: a gentle, modern introduction" by Stefanie Frank Singer, which bridges the gap between standard college courses on classical point mechanics and books like those mentioned above.

Another interesting book for geometrical hamilton mechanics is "Introduction to Symmetry and Mechanics" by J. Marsden, which gives a nice introductory overview to that topic.

For more examples from a geometrical point of view you may also consult "Global aspects of classical integrable systems" by Cushman and Bates.

One sould also mention "Mathematical Aspects of Classical and Celestical Mechanics" by Arnold, Kozlov and Neishtadt.

For a less advanced (and less rigorous approach) with very much examples you may have a look at the german book "Klassische Mechanik" by F. Kuypers.

Although not specifically answering the needs of Cedric H. (symplectic geometry) I find Introduction to Dynamics by Percival & Richards one of the best (and simplest) introductions to Lagrangian and Hamiltonian dynamics, particularly canonical transformations and so on. I point out this book because it is probably not so well known.

This isn't explicitly about just mechanics as it tries to hit a lot of different areas in physics but it covers the material you are asking for:

http://www.amazon.com/Differential-Forms-Applications-Physical-Sciences/dp/0486661695

It will absolutely provide you with a firm basis in the geometrical considerations you mentioned in your question (tangent bundle, cotangent bundle, 1-form, 2-form, etc.).

Classical Mechanics: Systems of Particles and Hamiltonian Dynamics by Walter Greiner.

This is a very good book for the same reasons that all the books belonging to the series of books written by Greiner are good. They are clear, they do not shy away from mathematics (they are written for people who want to pursue theoretical physics) and they have many examples.