For example, about the basic differential Calculus, I'm doing it in the following way, and is it the good method ?
I only focus and try to obtain the fully understanding and intuitive sense of the most basic and fundamental concepts or theorem, but for others things which could be proved or just 'special case aspect' of the basic concepts, I only to referring them when I use it, but not to try to understand or memorize.
Like, The concepts of Limits and Derivative and Theorem of Differentiability implies Continuity are the three most fundamental stuffs. I'll put focus on that to fully understand why it's true.
But for the rules, like $f'(x^n)=nx^{n-1}$, or sum rule, product rule, quotient rule and so on, I only know they are proved from the basic concept of derivative, i.e. $\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$. All the rules are from this basic concept. So I understand this basic concept intuitively means we find the instantaneously changing rates with respect to $x$.
But for the rules, like integer power differentiation, i.e. $f'(x^n)=nx^{n-1}$, I cannot 'understand' it in the same way with that of basic concept of derivative(instantaneously changing rates with respect to $x$). Only thing I know is that it's proved from the definition of derivative.
So, is it enough for mastering the rules ? Because for definition of derivative, I could still graph a function, and to see the trending when $\Delta x \rightarrow 0$, the limit will get the instantaneous changing rate at the point of $x_0$. But for the proved rules, like $f'(x^n)=nx^{n-1}$, it's very hard to get the intuitively feeling in the same way with the derivative.
Learn the history of Analysis, Specially History of Analysis from 16th century to Euler and Cauchy, Weistrauss School and Cantor. Analysis by history and History Of mathematics by John Stillwell. That will instill the sense of motivation for the ideas.