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, 

<blockquote>
Don't disparage the "rules" and computational aspects of differential equations. 
</blockquote>

*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:

<ul> 
<li> Piskunov, <em>Differential and integral calculus</em>
<li> Filippov, <em>Problems in differential equations</em>
<li> Arnold, <em>Ordinary differential equations</em>
<li> Poincaré, <em>On curves defined by differential equations</em>
<li> Arnold, <em>Geometric theory of differential equations</em>
<li> Arnold, <em>Mathematical methods of classical mechanics</em>
</ul>

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