Maybe I am reading too much into your pseudonym and your partly apologetic and partly condescending comments about the course you are going to take, but please,

Don't disparage the "rules" and computational aspects of differential equations.

Firstly, it is a beautiful subject with direct scientific origin and arguably most applications (save only calculus, perhaps) of all the courses you'd ever take. Secondly, these scientific connections continue to motivate and shape the development of the subject. Thirdly, rigor and abstraction are not substitutes for the actual mathematical content. Bourbaki never wrote a volume on differential equations, and the reason, I think, is that the subject is too content-rich to be amenable to axiomatic treatment. Finally, I've taught students who were gung-ho about rigorous real analysis, Rudin style, but couldn't compute the Taylor expansion of $\sqrt{1+x^3}.$ Knowing that the Riemann-Hilbert correspondence is an equivalence of triangulated categories may feel empowering, but as a matter of technique, it is mere stardust compared with the power of being able to compute the monodromy of a Fuchsian differential equation by hand.

Having forewarned you, here are my favorite introductory books on differential equations, all eminently suitable for self-study:

• Piskunov, Differential and integral calculus
• Filippov, Problems in differential equations
• Arnold, Ordinary differential equations
• Poincaré, On curves defined by differential equations
• Arnold, Geometric theory of differential equations
• Arnold, Mathematical methods of classical mechanics

You will find a lot of geometry, including an excellent exposition of calculus on manifolds, in the right context, in Arnold's Mathematical methods.