Since you majored in CS, you probably took calculus 1–3, a statistics and maybe a discrete math course at the minimum.

The first thing you need to do is to get your linear algebra up to speed. You should find a strong undergraduate text in linear algebra such as a Strang and then find a higher level, more theoretical book on linear algebra book. Every single topic in applied mathematics is linear algebra or it is gibberish. It is impossible to separate the two. This is especially true in optimization.

After that, most topics in applied mathematics will essentially be optimization, differential equations, or numerical analysis. Having a strong analysis background is essential. You should try to cover ordinary differential equations, partial differential equations, real analysis, complex analysis, and numerical analysis.

I can’t say that I’ve used topology at all in my applied mathematics courses. It is an interesting field, and, in fact, I’ve heard of numerical topology, but I imagine it is a fairly esoteric subject compared to the others I’ve mentioned.

Anyway, as I mentioned, if you get your linear algebra up to speed you will find the field will be very accessible to you.