I recently completed a solid chunk of my first truly formal math book on Set Theory and decided to move on to Michael Spivak's Calculus (as I heard it was a good place to begin one's journey into Analysis). For the purpose of self-studying, I purchased the solution manual, as well.
After completing the first two chapters and going through all practice problems, I am now painfully aware that I am not particularly great at proofs involving clever equation manipulations. As I progressed through my aforementioned set theory book, I felt like the number of "proof flavors" (different strategies one could invoke to tackle the claim) were quite manageable. Figuring out which path to pursue seemed to be categorically recognizable.
However, when it comes to proofs involving equation manipulation, I literally feel like I am staring into a sea of hundreds of possible permutations (in the English sense) that one might use to arrive at the desired conclusion. From what I have seen so far, the strategies largely seem to involve the invocation of Field Axioms (e.g. creative usages of $0$ and $1$ through the existence of additive inverses and multiplicative inverses, respectively) and factoring (that often times feels quite contrived).
I was wondering if there were any general tips and tricks as to how one might improve his/her ability in recognizing the type of equation manipulation that a given proof is suggesting (yelling) to carry out. I am assuming the seasoned mathsters have some sort of pattern recognition that goes off in their heads that allows for efficient determination of starting points.
I apologize if this question is a little too broad. I recognize that every proof may necessarily be "special", so in the absence of a specific example, perhaps this question is unanswerable. But here's hoping. Book recommendations are also greatly appreciated! Cheers~