1. You most likely wouldn't have had a problem solver's education. So the first thing to do would be do start understanding how problem solvers approach Mathematics. I suggest two books:
- The Art and Craft of Problem Solving by Paul Zeitz
- How to Solve It by George Polya
There are also others, e.g., http://www.amazon.com/Math-Olympiad-Resources/lm/1SWDJ5NA047...
Keep in mind that most proofs you see are highly polished for the sake of clear presentation. The route Mathematicians take to reach the proofs are almost always messy. And, unfortunately, they like to clear their tracks.
A great way to learn how to solve problems is to find a good problem solver and asking them to think out loud while they are solving problems. Also, try to find alternative proofs of any proof you learn.
2. Understand the history behind what you're studying. How did the ideas you're trying to learn (Abstract Algebra, Topology, Vector Analysis, etc.) come about? This will add an immense amount of meaning to your subjects.
I recommend the following books:
- "Men of Mathematics" by ET Bell. Also "The Development of Mathematics" by the same author.
- "A Concise History of Mathematics" by Dirk Struik,
- "A History of Mathematics" by Boyer
- And of course, the internet.
3. Read a few "big picture" books side-by-side. A few suggestions:
- "Foundations and Fundamental Concepts of Mathematics" by Howard Eves
- "Concepts of Modern Mathematics" by Ian Stewart
4. IMHO - and this is just my preference - study application of Mathematics to Physics and Engineering. Physics especially was an inspiration for a lot of Mathematics, and in addition, this will also let you solve concrete problems using the tools you learn.
5. During my undergraduate and graduate periods, I found several Schaum's and Dover books helpful. They're usually short and pretty cheap.
6. Initially, do not let rigor get in the way of understanding the content. Fourier, Newton, Euler, etc. weren't all that rigorous by modern standards.
Two more must read books:
1. "The Mathematical Experience" by Davis and Hersh (http://en.wikipedia.org/wiki/The_Mathematical_Experience#cit...)
2. "I Want To Be a Mathematician" (http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...).
Probably the key thing for me in helping my math ability was to actively try to prove theorems. Before reading a proof, I always try to solve the theorem myself first. And then after reading the actual proof... try to prove it again. You'll be surprised how many times you can't prove a theorem for which you just read the proof!!!
But this will help you get better at doing proofs, and understanding math. And it will also help you appreciate good proofs, because you would have already tried to solve it. You'll say, "Ahh... I didn't even think to try that, but that was exactly the step I would have needed".
Lastly, as someone else mentioned -- the proofs you read in texts are polished proofs. Often those theorems proved have been attempted by famous mathematicians who failed to prove it in their lifetimes. Take your time, be rigorous, and thoughtful. If you do that, you come out ahead regardless.
Can I ask you a more detailed question? Can you suggest resources to tackle machine learning and statistical inference of patterns in data? I am currently going through (http://www.amazon.com/Data-Mining-Techniques-Implementations...) and there is so much depth in it that I want to understand. This is why I wanted to strengthen my fundamentals first before moving on to the fancy stuff.
If you're in this field or have any idea of it can you suggest me a gentle introduction to the depths of it?