Reddit mentions of Mechanics: Volume 1 (Course of Theoretical Physics S)

Landau and lifshitz mechanics is short enough.
http://www.amazon.com/Mechanics-Third-Edition-Theoretical-Physics/dp/0750628960

Any mechanics text targeted for the standard junior level mechanics course for majors will cover it. I used Fowles and Cassiday when I took it. I'm not really sure what else is standard. The standard text in grad courses is Goldstein, which should be approachable by an undergrad at least. If you're crazy and a classical mechanics junkie like I was as an undergrad, Landau and Lifshitz vol1 is a beautiful treatment (that you unfortunately probably already need to have seen the material once to appreciate. Oh well. Like I said: if you're crazy). The issue here is that sometimes undergrad courses will skip these (as I learned, amazed, when I was encountering other grad students that hadn't done Lagrangian mechanics before) so make sure you read those chapters and do the problems: quantum mechanics is done in a hamiltonian formulation, and quantum field theory in a Lagrangian formulation (the latter is because the Lagriangian treatment is automatically relativistici)

I never had a course specifically on waves. It's something you'll likely hit pretty well in whatever non-freshman E&M course you take. Beware though that some courses targeted at engineers will do AC circuits at the expense of waves. But the text is still useable to look into it yourself.

Brush up on mathematical methods for physics. Learn Linear Algebra, Ordinary and Partial Differential Equations, Multivariable Calculus, Complex Analysis, and Tensor Analysis. A good book would be this: http://www.amazon.com/Mathematical-Methods-Physical-Sciences-Mary/dp/0471198269/ref=ntt_at_ep_dpi_1

Classical Mechanics: http://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960/ref=sr_1_7?s=books&ie=UTF8&qid=1291625026&sr=1-7

E&M: http://www.amazon.com/Electromagnetic-Fields-Roald-K-Wangsness/dp/0471811866/ref=ntt_at_ep_dpi_1
or http://www.amazon.com/Introduction-Electrodynamics-3rd-David-Griffiths/dp/013805326X/ref=sr_1_1?s=books&ie=UTF8&qid=1291625100&sr=1-1

Statistical Mechanics: http://www.amazon.com/Fundamentals-Statistical-Thermal-Physics-Frederick/dp/1577666127/ref=sr_1_1?ie=UTF8&s=books&qid=1291625184&sr=1-1

Quantum Mechanics: http://www.amazon.com/Principles-Quantum-Mechanics-R-Shankar/dp/0306447908/ref=sr_1_4?s=books&ie=UTF8&qid=1291625261&sr=1-4

As with most things you gotta know the basics. Start with classical mechanics. The best book is Landau's Mechanics, but it's quite advanced. The undergraduate text I used at university was Thornton and Marion. If that's still too much I've heard Taylor's book is even gentler.

Also, make sure you know your calculus.

For Variational Calculus, the best references are Landau and Lifchitz and Gelfand and Fomin. The former is really a mechanics book that incorporates variational calculus in a very rigorous manner that one would expect from a theoretical physicist. The latter is a straight-up variational calculus book. Both are relatively cheap (you can find landau for cheaper than the amazon price).

For non-commutative geometry, there is this classic paper. /u/hopffiber gave the classic references for the rest of the topics, although you should think about learning quantum field theory since all the applications of Lie algebras come from QFT and String Theory. There are some excellent notes by David Tong that you can find with google-fu.

My recommendations:

• The Feynman Lectures on Physics - a grand, insightful introductory (i.e. first year college) course from the master. I wish I had read this in high school.
  
• Mechanics by Landau and Lifshitz - volume I of the famous course on theoretical physics
  
• Structure and Interpretation of Classical Mechanics - if you are also a programming nerd
  
  See also:
  
  
• http://math.ucr.edu/home/baez/books.html
  
• http://physics.stackexchange.com/questions/tagged/books
  

A solid intro book to QM is Zetilli, but as others have mentioned you might want to learn some Classical Mechanics first and for that I recommend Landau or Goldstein. Landau is usually more of a grad book and Goldstein is an undergrad one.

Thanks. The HJE is usually included in a course on advanced classical mechanics. Landau and Lifshitz do a great job with it, but I actually prefer a more direct derivation.