EDIT: As Vladimir Nesov points out below, don't just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.

Also refined point (7).

I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I'd recommend, coming from different goal structures I have:

1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher's visual approach. This doesn't require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.

2) if you are interested in studying advanced mathematics, I'd recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians' perspectives on category theory seems valuable to me. Also complex analysis.

3) If you are interested in abstract concepts that feel like they have universal applicability (I don't know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.

4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I'm not really sure.

5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.

6) You probably shouldn't study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it's appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!

7) if you are in school at a university, I'd suggest looking up math professors on ratemyprofessors, the ratings aren't perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors' teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won't. I don't think this is the context of your question but it is probably relevant to others asking similar questions.

I am interested in Computer Science, compiler optimization, and machine learning but I know relatively little about these subjects.

I am currently reading Concrete Mathematics.

I am having trouble understanding this question. By the words "mathematical concentrations", could one substitute the term "mathematical subfields"

Yes, thank you, I performed the substitution in the original post.

If so, then I would say the best resource could be more experienced people with similar instrumental goals to one's self.

Which is precisely why I am asking LessWrong.

What resources exist detailing which mathematics to learn in what order?

One could acquire a university syllabus for a degree in mathematics and use textbooks and or MIT Opencoursware to learn with.

What resources exist that explain the utility of different mathematical concentrations for the purpose of directing studies?

I am having trouble understanding this question. By the words "mathematical concentrations", could one substitute the term "mathematical subfields"? If so, then I would say the best resource could be more experienced people with similar instrumental goals to one's self.

I listen to a weekly podcast called skeptoid. It's a very concise, information packed show focusing on pseudoscience and rationality. I've listened to episodes on the validity of IQ testing all the way to an entire episode on Zeno's paradoxes.

It may be light-hearted and entertaining, but it's still a very informative and educational podcast.