I'm on math carreer and I'm finding it very difficult, I've read "How to do a demonstrarion" and "logic for math" books, but I think I could really use a book full of things like "How to prove that a set is dense in another set", "How to prove this recursive function is bounded", "How to use the Stone-Weierstrass theorem".
In the first year of my studies I found
How to read and do proofs an introduction to mathematical thought processes
It really helped me understand the basics.
However, to learn the application of theorems, like Stone-Weierstrass, one has to see those in action. I don't think there is a book that incorporates strategies for every theorem one needs.
In any ordinary book on calculus there should be plenty of examples how to use those theorems. Reading those textbooks can improve our "intuition" on how to use theorems.
Finally remember this quote form Euclid:
There is no royal road to geometry.