Reddit mentions of Ordinary Differential Equations (The MIT Press)

This is a really popular theoretical differential equations book http://www.amazon.com/Ordinary-Differential-Equations-V-I-Arnold/dp/0262510189

It's Ordinary Differential Equations by V.I. Arnold, it's highly regarded and I see people recommend it over on math.stackexchange all the time.

However I'm not sure if it's the kind of book you're looking for because I don't believe it's an introductory book at all. From what I've heard it's pretty advanced.

Hopefully someone more knowledgeable than I can explain whether this book is appropriate for you or not.

I think learning proofs-based calculus and linear algebra are solid places to start. To complete the trifecta, look into Arnold for a more proofy differential equations course.

After that, my suggestions are Rudin and, to build on your CS background, Sipser. These are very standard references, though Rudin's a slightly controversial suggestion because he's notorious for being terse. I say, go ahead and try it, you might find you like it.

As for names of fields to look into: Real Analysis, Complex Analysis, Abstract Algebra, Topology, and Differential Geometry mostly partition the field of mathematics with corresponding undergraduate courses. As for computer science, look into Algorithmic Analysis and Computational Complexity (sometimes sold as a single course called Theory of Computation).

Smooth manifolds play an important role in the theory of dynamical systems, so I would suggest learning that after, or concurrently with smooth manifolds. There's a good book by V. I. Arnol'd.

• topology: Introduction to Topology and Modern Analysis by G. Simmons
  
• real analysis: A First Course in Real Analysis by Protter and Morrey
  
• complex analysis: Theory of Functions of a Complex Variable by A.I. Markushevich
  
• differential equations: Ordinary Differential Equations by V.I. Arnold
  
• linear algebra: Linear Algebra by G. Shilov
  
• calculus: Calculus with Applications and Computing (vol.1) by P. Lax
  
• probability: Introduction to Probability Theory by Hoel, Port and Stone

The book for my undergrad diff eqs class. I highly recommend it if you have an introductory background in ODEs, but even if you don't (I didn't going in), it's a great book.

Wow, ambitious! I'd highly recommend V.I. Arnold's book on ODEs: http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 ... not only is it a great book in itself, but it should give you an excellent foundation for differential geometry and more advanced geometric mechanics (e.g., Lagrangian/Hamiltonian mechanics, dynamical systems, etc.).

What differential equations book would you use? Some are just plug-and-chug like yobyeknom said, and others are highly conceptual (like this and this).

A plug-and-chug course probably wouldn't require much in the way of a background in linear algebra. The latter certainly would.

Why don't you ask your advisor? They would know better than us.