I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$.
But I feel as though I'm missing some important implications when we state that $f$ is in $L^p$. What else is implied in the definition of $L^p$?
Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. If $X$ is a set, $\mathcal{S}$ a sigma algebra on $X$ and $\mu : \mathcal{S} \to \mathbb{R}^+$ a measure, then, by definition $$ L^p(X, \mathcal{S}, \mu) = \left\{ f:X\to \mathbb{F} : f \text{ is $\mathcal{S}$ measurable and } \int_X |f|^p d\mu < \infty \right\} $$ In that case, if $f \in L^p(X, \mathcal{S}, \mu)$, the $p$-semi norm is $$ \| f \|_p =\left( \int_X |f|^p d\mu \right)^{1/p} < \infty $$ Note that you forgot to consider the modulus $|f|$ in the semi norm