Geometrically Speaking:
The dot product is vital for measuring lengths and angles. First ${\bf u} \cdot {\bf u} = ||{\bf u}||^2$ and second, assuming ${\bf u}$ and ${\bf v}$ are non-zero, we have ${\bf u} \cdot {\bf v} = 0$ if and only if ${\bf u}$ and ${\bf v}$ are perpendicular.
The cross product of two non-zero, non-parallel vectors gives a third vector that is perpendicular to the first two. Thus: ${\bf u} \perp ({\bf u} \times {\bf v}) \perp {\bf v}.$
Moreover, the dot product and the cross product are related by the triple scalar product:
$$[{\bf u},{\bf v},{\bf w}] = ({\bf u} \times {\bf v})\cdot {\bf w}$$
The triple scalar product is the volume of the parallelepiped spanned by ${\bf u}$, ${\bf v}$ and ${\bf w}$.
Abstractly Speaking:
The dot product can be used to show that a vector space $V$ is isomorphic to its dual vector space $V^*$. Recall that $V^*$ consists of all linear maps $f : V \to \mathbb{R}.$ The isomorphism $\phi : V \to V^*$ is given by $\phi : {\bf u} \mapsto f_{{\bf u}}$ where $f_{{\bf u}}({\bf v}) = {\bf u} \cdot {\bf v}$ for all ${\bf v}$ in $V$.
TL;DR: Learn by reading, doing, following online courses. Join groups/clubs. Work on projects. Find the area you really enjoy. Eventually you'll contribute to helping people out especially if you believe in whatever field you end up joining.
I love your awesome attitude about math and I wish there were more people like you. Well, there is a lot you can do to help people with math. A LOT! It really boils down to what do you want the world to be like and help it be more like that.
Do you want a greener/cleaner world? Go into environmental engineering.
Do you want a healthier world? Go into the biomedical sector.
Do you want a more technological advanced world? Go into computer science or some technology related engineering field.
Do you want people to be more educated? Go into teaching.
Do you want to do either of these or more but the current environment won't let you? Start a startup company.
I could go on and on and on with these recommendations. The thing is that mathematics is a very flexible subject. You can do with it whatever you want. Heck, there are even Simpsons and Futurama writers who are mathematicians: http://en.wikipedia.org/wiki/Jeff_Westbrook
Don't worry too much about helping people out. It kinda comes as a side effect of doing math. For example, studying graph theory back in the days of Euler may have sounded niche and a waste of time but if it were not for graph theory, no internet. (and certainly no facebook... although... i know people who think that wouldn't be such a bad thing)
Focus on finding something you really like and just roll with it. My recommendation... read, read, read. If you have difficulties taking those courses, read the books. A trick I used to do is that I looked at the syllabi of the courses I wish I could take and just got the textbooks and went through them chapter by chapter.
Learn as much as you can: set theory, graph theory, game theory, optimization, cryptography, differential equations, algebra... whatever you can. And try not to limit yourself to mathematics. We live in a complex interconnected world. Many ideas in math today come from economics and biology and physics and all sorts of places.
When you say you'd like something like taking classes... you know, there just isn't enough time and it is not always possible. But youtube exists. And all those MOOCs (Massive Online Open Courses) like Coursera, EDX, or Udacity. They have loads of courses, some more mathy than others. Even all those computer science Udacity courses are interesting because computer science in the end is all about math.
But learning is not enough, actually working on projects is way more helpful. If you have access to clubs or can get friends to work on some project, you'll learn so much more. Just a simple project like getting a little robot to follow a line on the floor or making a video game will challenge you mathematically and thus you'll get to see how you can apply math in the real world.
So, don't worry too much. Just by your attitude I can tell that you'll manage to do good in the world with math. Just learn, practice, and make awesome.
Best Answer
Daniel, I understand what you mean. Sometimes it's difficult to understand abstract proofs, especially when the proofs only involve many variables/symbols/greek letters and not numbers.
What usually works for me is to get 1 or 2 examples (either from books, online, lectures, etc.) which illustrate how the theorem works. I feel that this "concreteness" helps me understand what is going on better, and then I go back and re-read the proof the of theorem. Usually easy, trivial examples are best.
Then, when I've understood the theorem and its proof well, I can try to apply it to more difficult examples.
So for example in Linear Algebra, when learning the Rank-Nullity Theorem, you could find a very simple example (maybe 2x2 matrices) from a textbook first to see it in practice. And perhaps try another yourself to see the Rank-Nullity Theorem in practice. Then afterwards go back and re-read the proof.
Hope that helps.