How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up.
EDIT: Could anyone suggest books for someone with a math education that just barely touches on high-school Algebra (got into parabolas, rationalizing, some graphing and functions). This is what I am currently doing: attending high school as a Junior.
On
Be honest with yourself about what you do and don't understand. Don't fall victim to "proof by intimidation," where someone attempts to shame you into saying you understand something by implying that you're dumb if you don't. Always ask questions until you really get it. Similarly, don't let yourself move on before you understand something fully; pretty much everything will come back to bite you at some point down the line. If this seems like a pessimistic attitude, it's not - it is simply humility, and humility is the path to genuine knowledge.
EDIT: I just noticed your question about books. How To Prove It is a great transition to more advanced mathematics. After that, check out some of the Dover books, they are all very cheap and most are decent introductions to their respective subjects.
On
I would quote this excellent article by Peter Norvig.
It's about programming, but applies to all other domains as well.
Researchers have shown it takes about ten years to develop expertise in any of a wide variety of areas, including chess playing, music composition, telegraph operation, painting, piano playing, swimming, tennis, and research in neuropsychology and topology.
The key is deliberative practice: not just doing it again and again, but challenging yourself with a task that is just beyond your current ability, trying it, analyzing your performance while and after doing it, and correcting any mistakes. Then repeat. And repeat again.
On
If you're looking for a place to start, pick a video that covers something that you already know from this list of PatrickJMT videos, watch it, and then watch the videos after that in order. Try coming up with problems, and solutions, yourself. You could also try googling the name of the problem with "practice question" or "quiz" to get some premade questions.
This is also an excellent resource if you come across a problem and don't know how to proceed. Always try to solve the question after getting some insight before watching the whole video and 'copying' the answer.
Good luck!
On
Just try to learn about proofs and logic. If you have difficulty with proofs this book is a good way to start: How to Prove It: A Structured Approach by Daniel J. Velleman
On
There is a nice book by Polya "How to solve it". It does not teach particular mathematics but rather "a way of mathematics". It's also quite elementary.
On
You may find it useful to start with http://www.khanacademy.org/
Once completed, look for additional resources and side by side track how much you completed from http://mathworld.wolfram.com/
On
Alexander Gruber's suggestions are excellent. Definitely take them on board. I'd also counsel patience and an appreciation for mathematical elegance. It's often the case that once you understand a concept, you're surprised at how simple and concise it really is. But don't rush to understand it. Enjoy the process of learning.
There's a fair amount of research into the connection between emotion and learning (e.g. http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.elsevier-9bf96239-1b71-30a0-a875-8878e78f28c4/c/main.pdf), so don't neglect that emotional connection. Most mathematicians have it to a greater or lesser degree.
On
One thing that I cherish myself is I ask myself after learning a new topic or subject for a class is: Do I really understand everything that is going on?
If you don't understand everything going on you might struggle mightily with an application of the topic or further work in the course relying heavily on that specific topic.
If you don't think you understand it well then go ahead and write down what questions you have about the specific topic and seek answers. Whether it be MSE, your textbook, or your professors, you are likely to find the answers to those questions $somewhere$.
Not only will this help you in your current studies, it will help further your cementation of mathematics in your mind, which is always good because mathematics is a field that builds upon itself in many ways. (For example, If you struggle doing arithmetic, you are likely to struggle in courses that use it heavily.)
Buy a huge whiteboard and think of math like a puzzle. Hours can pass by quick with space to scribble and self-motivation. Mathematics has to become a hobby for you to actually come to understand it. Know your basics well, even if it takes a while longer than expected to have a solid standing.
And in terms of books, check Amazon. I surf the web if I've got a specific topic in mind though.