What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?
A classic book is Coddington & Levinson, Theory of Ordinary Differential Equations. Probably tough going for most students.
Coddington, An Introduction to Ordinary Differential Equations. Much shorter, suited to undergraduates, but still rigorous. There is an inexpensive reprinting from Dover.
There's nothing wrong with learning it rigorously, but I would recommend to learn it "non-rigorously" first and then rigorously. E.g. in the context of a "graduate" ODE course. That's because there are some very important aspects of the topic that you may miss if you only concentrate on rigor, first. Like an intuitive understanding of the 2nd order ODE and the whole forcing function, undamped, overdamped, etc. And the sources that concentrate on rigor often assume that you have the non-rigorous understanding...not just that it's harder without the background, but that you never learn some key insights.
In that vein, a couple texts that I like that will still give you the basics and are slightly rigorous are Murray Spiegel Applied Differential Equations (3rd edition) and Ordinary Differential Equations Tenenbaum and Pollard. After that, I would move to a graduate ODE text, probably something in Springer, for the theory.
But really, you are missing a major trick if you only consider ODEs (or worse PDEs) in the context of analysis/topo, with zero physical insights or references to the rich science and engineering applications.
There are many, many books on ODEs, many of them good.
For the basic theory no one seems to have improved much on the book of Coddington and Levinson cited by Gerald Edgar. That book is old-fashioned, has essentially no examples, and could be seen as quite dry. It is clearly written and I was able to learn from it.
V. I. Arnold. Ordinary Differential Equations is an excellent book because the geometric ideas behind ODEs are explained. Arnold does not write out all the details (for me this is a virtue of his books), but the writing is always careful and one learns a lot by filling in the details. His more advanced book Geometrical methods in the theory of ordinary differential equations is wonderful but requires some background.
Another book in which topological/geometrical ideas operate is that of M. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra.
A student who has a good background in geometry and topology should also learn about numerics and practical solvability issues.
For learning the theory behind numerical methods for solving ODEs I like John Butcher's Numerical methods for ordinary differential equations.
A good introduction to how ODEs are solved in practice is the self-descriptively titled Hairer, Nørsett, Wanner, Solving ordinary differential equations. I.
For a student with geometric interests it might be interesting to read Numerical Hamiltonian problems by Sanz-Serna and Calvo and Geometric numerical integration by Hairer, Lubich, and Wanner. These are about numeric methods adapted to the underlying geometry related to the flow being integrated.
I'm not an expert in ODEs and as I observed there are many, many books available. I just listed some I like written by people with good mathematical judgment.