In my self study of calculus, I've found that there are examples in the books i read where the author rewrites an equation or expression either as part of a logical step in a proof, or to simplify it so that he can perform other desired operations on it.
But when it comes time for me to try practice questions, i look at the expression/equation and have no intuition or idea as to how to meaningfully simplify or rewrite it to suit my purpose. I also have a tendency of looking at expressions and initially getting intimidated by their complexity
Thus I'm interested in studying a field which will help me become comfortable with working with, manipulating and rewriting expressions/equations including basic operators, exponents, log and trig functions.
From my limited understanding basic algebra would be a good starting place, but which more advanced fields should i study to get an even stronger/more advanced ability to be comfortable with/manipulate expressions and equations?
In my opinion one learns those kind of manipulation by experience and i don't think that there is any area which do not uses such things. Consider for example that you need to prove $|x-y|\leq |x-z|+|z-y|\ \forall\ x,y,z\in \mathbb{R}$. The standard trick is to write $x-y=x-z+z-y$ in LHS and then proceed using the properties of modulus. Now a person who sees it for the first time may wonder that how one manipulates such thing but slowly while doing similar kind of operations one gets accustomed to it.