Provided your student has a strength, you can play to that. In some sense that's what finding "applications" is about -- appealing to someone's immediate values. But there's also a danger in that, if all you do is appeal to someone's immediate values you may be forced to build up a rather contrived and limited notion of what mathematics is.
So in teaching it's also very useful to take students to foreign territory where they're weak, and build up their appreciation for what they may have found useless or "abstract". Finding applications isn't the only way to do this -- simply providing re-interpretations of things they know that seem complicated can be useful. If something they found hard becomes easier in a different setting, this will frequently be perceived as useful.
I'm teaching an introductory calculus class this semester. In the last few classes we've been differentiating trig functions. Recently students have been asking questions like "can we have a formula sheet on the exams?" The bulk of the number of things to remember is starting to appear heavy. Their concern is that things like the double-angle formula or the formula for the derivative of $\csc(x)$ are difficult to remember. So I spent a little bit of time on how you can get by, not remembering much at all about trig functions.
Here's the pencil-sketch of the idea that you can flesh-out:
I take it as an axiom people know what the graphs of $\sin(x)$ and $\cos(x)$ look like -- and students should be able to deduce these things from "$(\cos(x),\sin(x))$ is the point on the unit circle that you get to if you lay-out a string of length $x$ along the unit circle, starting from $(1,0)$ and going in a counter-clockwise direction".
So how do you remember what $\sin'(x)$ is? You sketch the graph of $\sin(x)$, lay down several tangent lines, and plot the slopes of the tangent lines. Observe that the slopes vary continuously, and fill in the gaps appropriately with your knowledge of the local maxima and minimal of $\sin(x)$. Then you observe the graph looks like $\cos(x)$, so you conclude $\sin'(x)$ must be $\cos(x)$. This isn't a formal conclusion, more of a seat-of-the-pants "well, how complicated can $\sin'(x)$ be?" type argument.
Similarly, the double-angle formula for $\cos(2x)$. How do you remember that? The idea is that we know there is a formula that relates $\cos(2x)$ to either $\cos(x)$ or $\sin(x)$ but people tend not to have the exact form on immediate recall. I sketch the graph of $\cos(x)$, then sketch the graph of $\cos(2x)$ as the previous graph with double the frequency. $\cos(2x)$ has twice the frequency -- you can also get twice the frequency by looking at $\cos^2(x)$ or $\sin^2(x)$ -- so spend some time sketching the graph of those functions. This amounts to knowing the graph of $\sin(x)$ and $\cos(x)$, plus the graph of $x^2$ -- to know that $\cos^2(x)$ for example not only has twice the frequency but that when $\cos^2(x)=0$ the function looks quadratic -- like $x^2$ around those points. From there it's just a scaling argument to say $2\cos^2(x)$ looks like $\cos(2x)+1$.
So the point of all this is that, with enough patience and geometric insight, all these formulas that students encounter you can cast them as things that, by their very nature they must be easily re-discoverable with very little probing. In particular, provided you're comfortable enough graphing functions it's less energy (and really far more informative) to remember the forms of these identities by intuiting them from your rough expectations. In that way you can re-cast identities like
$$\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$$
in a different light. The fact that this identity has such a simple form is somewhat remarkable -- but once you know it has a simple form, re-discovering it via non-rigorous arguments and heuristics is rather immediate.
Certainly, this line of reasoning isn't great for all students as it assumes a certain familiarity and an intuition with graphing that many students may not have. But once students have the knowledge that this is something they can think about, and that other people think this way -- that the discovery process is full of intuition and guesses before anything rigorous -- this sets them on a path.
The Math Subject GRE is 50% Calc 1, 2, 3, and Differential Equations. High school algebra and linear algebra are another 15-20% probably. If you do well on just those questions, you will be in the 70th or 80th percentile. Note, this is compared to students wanting to study math at graduate school, so this is very good. Learning several entirely new subject will take months, probably more time than you have, and will end up adding just a few points to your score. Mastering the subjects you have already had will help your score much more.
Here is a link to a previous test, including the breakdown of subjects.
http://www.ets.org/Media/Tests/GRE/pdf/Math.pdf
Note 25% is "Additional Topics". You'd need to learn several semester courses worth of material to get this stuff. Don't worry about that.
Best Answer
While studying analysis/calculus, I've found the following book to be quite interesting/motivating, even though I'm not so sure you can master the subject just by reading it alone:
Analysis by Its History, by Ernst Hairer and Gerhard Wanner.