How to use Advanced Algebra II
Free educational resource, available under a Creative Commons License. I'm not sure if this is what you are looking for, but it's free, so it can't hurt to take a look.
Also, I used Schaum's Outlines for self-study of Probability and Statistics. I liked it and they have ~60 books in the mathematics category. McGraw-Hill Professional - Mathematics (They had a couple of those at my library, and I'm Dutch, so if you're English you can probably find some in yours too.)
Practice makes perfect. Do a lot of exercises until you feel you get it.How do other people usually develop this capacity?
Finally, I'd like to add that you actually are explaining yourself very well. What you're describing is more or less what I personally feel is mathematics: The puzzle of combining various rules/methods to solve a problem.