There are simply too many vector identities including $\nabla$. Every time I read a physics book I must learn some new ones. For example, $$ \nabla\times (\nabla\times \mathbf f)=\nabla(\nabla\cdot \mathbf f)-\nabla^2\mathbf f,\\ \nabla(\mathbf{A} \cdot \mathbf{B}) \ =\ (\mathbf{A} {\cdot} \nabla)\mathbf{B} \,+\, (\mathbf{B}\, {\cdot} \nabla)\mathbf{A} \,+\, \mathbf{A} {\times} (\nabla {\times} \mathbf{B}) \,+\, \mathbf{B} {\times} (\nabla {\times} \mathbf{A}). $$ Many of such identities contain four terms on RHS (unlike the two terms for chain rule of differentiation), and it is very difficult to remember the exact form of each of them.
Is there a fast way to understand many vector calculus identities quickly and intuitively? What should I learn first and what can be derived by myself?