Developing Mathematic Intuition I'm an engineering student, currently working my way through the fundamental mathematics courses.
I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, Pre-Calculus, and Calculus I (I'm currently struggling quite a bit in Calculus II; so only time (and sweat; no blood or tears yet) will tell if I can maintain my academic performance after this course.
However, although my school is good and well-ranked among community colleges, it's still a community college. None of the courses go too in-depth on any of the topics we cover. It's all about teaching us techniques and methods for solving problems (not extraordinarily difficult problems, either). It's not that the instructors aren't good - many are quite good and certainly know their math. But there just isn't time to spend on any individual topics. We covered all of the integration techniques that are taught at this level (with the exception of improper integrals) in about 2 weeks, or 8 class meetings.
In spite of this (or maybe because I've realized a lot of the responsibility for learning the rest falls on me), I've really developed an awe and a love for mathematics. Not enough too switch majors; I still have an overwhelming desire to build robots. ;) 
But I really want to master the subjects in mathematics I'm exposed to, to really learn them thoroughly and at a deep level—not only because the better I do that, the better an engineer I'll be (I hope), but also because I'm really blown away by how cool the math is.
So, my question is, how can I develop more adept mathematic thinking and reasoning skills, better math intuition?
None of my classes have been proof-based, yet. Would starting to learn how to build proofs help my intuitive skills to grow faster?
For instance, I've been studying (and struggling with a lot) infinite sequences and series, and how to represent functions as power, taylor, and maclaurin series. 
I have made some progress, but I'm advancing very slowly. When I look at a formula like:
$$P_0(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n} n!^2}$$
or even a more simple one, like:
$$\sum_{n=1}^{\infty} \frac{(-1)^n 3^{n-1}}{n!}$$
I have a great deal of trouble seeing past the jumble of variables and constants to the pattern they describe. I want to reach the point at which I can see the matrix!  ;)  (the movie type, not the spreadsheet type).
That's a joke of course, but seriously, while a mathematician may look at a matrix and see a mathematical structure, I have to think very hard, and sometimes to sketch an actual structure, to see a matrix as anything more than a large table of numbers.
If learning to prove theorems isn't the answer, (or the whole answer), what are some things you can try to help increase your capability to think mathematically / logically about concepts in calculus, and mathematics in general?
 A: One of my teachers always told me "don't know definitions, don't know math." At the time I was pretty annoyed, but he was completely right. The only way to learn math is to have the fundamentals down cold. This involves both a rigorous side, (memorizing them is a good start) and an intuitive side. So at an entry level, I strongly recommend spending a long time with the definitions. Theorems are nice and can help you understand the relationship between the definitions. But as far as Intuition goes, don't dive into the mechanics of the theorems too early. 
Some big ones from calculus are limit, Taylor series, integral, derivative/differentiable, open/closed, even/odd, and continuous. If you know those you can probably talk to anyone about calculus. 
The only way to build your intuitive understanding is to fail. Getting it wrong is the first step to getting it not totally wrong. That means trying a lot. Do your homework carefully. Try to ask follow up questions. A good curriculum can help reduce the amount of time it takes, you'll have to be patient no matter what. Do examples. Do hard examples. Do more examples. Do counter examples. Do not just settle for "well, $0$ satisfies the equation so it's probably fine." We've all done that, but it's bad practice. 
You know you're on the right track when you can see why a definition was picked the way it was. That is the real heart of intuition for definitions. For example, why should the coefficients for Taylor series look like they are? What properties do we even want from a taylor's series? Well, polynomials are awesome and simple. So let's use polynomials to approximate stuff. Ok... but how can we pick good approximations? It turns out it has something to do with making the $n^{\text{th}}$ derivative have the right value. It's worth understand how that works.
It sounds like you're on the right track. Half the battle is wanting to do it. The other half is work. 
Also, this site is a good resource. Learning to ask good questions here will be super helpful for you. 
A: Intuition and logic are not the same thing. Take, for example, the idea that
$$\lim_{x\to\infty} \frac{1}{x}=0$$
What does this mean? Intuitively, you can imagine a graph of the function and see that it gets closer and closer to $0$, but who's to say that the limit isn't actually $0.0001$? To show that this isn't the case, you need a formal definition of what a limit actually is, and you need to logically prove that this function's limit fits that definition. Making the proof may seem less intuitive than simply observing that the function approaches $0$.
To build intuition, what you need to do is learn how to visualize. For example, the first time I was asked to determine whether $n^n$ or $n!$ would be larger, I imagined the expressions as
\begin{gather}
n^n=n\times n\times\ldots\times n\\
n!=1\times 2\times\ldots\times n
\end{gather}
Clearly, $n^n>n!$. After understanding the intuitive part of a concept, then you can prove it to verify your intuition. Visualization will help you understand your proofs because you get a sense of how to proceed with them. You can improve your visualization by looking at graphs and diagrams of concepts.
Some mathematicians would object to relying on intuition over logic, as some statements that appear true at first are actually false. But as a non-math student trying to understand subjects that have been studied before, you shouldn't be concerned with this.
I would recommend doing all your math only when you are absolutely sure that the concepts make sense. Try to intuitively understand what your books say, and if that fails look for a proof. When you find a proof, go through every step and make sure that you can see why one statement follows from the next. And finally, doing lots of problems doesn't hurt, because it exposes you to the intricacies of an idea. But don't waste your time doing problems that don't challenge you.
A: You could check my answer here.
A: Zach Stone and tommytwoeyes gave great answers. In addition I’d caution that you can’t go equally deep into all areas of mathematics, at least not without doing so over a very long period of time. This isn’t specific to math. You can either be a generalist (having shallow knowledge broadly), a specialist (having shallow knowledge broadly and deep knowledge in a narrow area). Trying to be a specialist in everything leads to failure in everything because you have a limited amount of time in your day, and any time you spend going deep into one area takes time away from something else. So realistically you’ll want to make sure you work as hard as needed to do well in school, and then if you want to go deeper, pick one or two narrow areas at a time that you want to really dive deep into, and work hard exploring those areas with the techniques noted in other answers.
