I was reading answers to How do you explain the concept of logarithm to a five year old? and realized that I really don't understand mathematics. Okay, I lied when I said that I had no mathematical background, but it is as bad as that. I did learn mathematics throughout my education (I am a graduate) as part of my curriculum, but either due to my lack of interest or due to poor instructors in this part of the continent, I never really quite understood it. I became a programmer and started liking mathematics. I mean I see some solutions on project euler and I am really fascinated by them. Can someone point me to a resource/book/approach which will teach me mathematics from grounds up with a focus on understanding what goes under the hood? Not just ability to solve problems, but an in general solid understanding of mathematics which will make appreciate the concepts and then I guess the problem solving will come slowly. I have no time frame to learn anything, I just want to really learn and understand mathematics.
Not easy in only one book...
I'll suggest an excellent and cheap 1120 pages book from Aleksandrov, Kolmogorov and Lavrent'ev 'Mathematics: Its Content, Methods and Meaning' (review at Amazon).
It contains no exercises but starts with pretty basic facts.
From my personal experience, I think it's best to find an entry point at a level that you can do and experience real math as it is done. You will get a sense as to how the thought process develops and keeps building on its base.
In perhaps an oversimplification, math depends on rigor, so at every turn you have to self-reflect as to whether, as you put it, you are really "under the hood."
And there is a leap from mechanics to a fundamental understanding of the math process.
As you cultivate that skill - is is usually referred to as "maturity" - you will be able to bring that to delve into other areas of curiosity.
So my suggestion is to, continuing your analogy, to get your hands dirty.
One book that is highly acclaimed and is a very good entry point to this process is linked:
Giving this a try, you will be in the game rather than a spectator.
Math is a world like no other; hope you like it.
"The Princeton Companion to Mathematics" covers a wide range of non-trivial mathematics. It might not be what you are looking for, but it might be worth finding a library copy to browse.
The Prime Number Theorem, by G. J. O. Jameson is an excellent book for self-study.
For analysis, I would recommend the book Calculus by Michael Spivak, but the exercises in that book can definitely be a bit difficult.
If you'd rather start with algebra rather than analysis, there is basically no better introductory book than Linear Algebra by Hoffman and Kunze