Here's a theorem I find useful:

Theorem. Let $\phi \colon X \to Y$ be a smooth morphism of schemes. Then there exists an open cover $X = \bigcup U_i$ of $X$ such that each $U_i \to Y$ factors as $$U_i \stackrel \pi \to \mathbb A^{d_i}_Y \to Y$$ for some $d_i \in \mathbb Z_{\geq 0}$, with $\pi$ étale. Mnemonic: smooth morphisms have étale coordinates.

See Tag 054L.

Here's a theorem I find useful:

Theorem. Let $\phi \colon X \to Y$ be a smooth morphism of schemes of relative dimension $d$. Then there exists an open cover $X = \bigcup U_i$ of $X$ such that each $U_i \to Y$ factors as $$U_i \stackrel \pi \to \mathbb A^d_Y \to Y,$$ with $\pi$ étale. Mnemonic: smooth morphisms have étale coordinates.

See Tag 054L.