I have a lot of trouble remembering how the Proj functor works and how to think about sheaves of $\mathcal{O}_X$-modules on these spaces. Part of what makes this hard is that everybody finds it technical and explains it in their own way; in this document, I'll write things down in my own way (lifted from Hartshorne but more explicit, probably) so that I have a reference.
Let $S$ be a graded ring, and let $\mathsf{Proj}(S)$ be the set of homogeneous primes in $S$. To topologise $\mathsf{Proj}$ we take the sets $V(\mathfrak{a})$ as our closed sets, and to get a structure sheaf we define

where $S_{(\mathfrak{p})}$ is the ring of degree 0 elements in $T^{-1}S$ and where $T$ is the multiplicative system of homog elements in $S \setminus \mathfrak{p}$. One then shows that this makes $\mathsf{Proj}(S)$ a locally ringed space, having distinguished open sets $D_+(f) = \{\mathfrak{p} \ | \ f \not \in \mathfrak{p} \}$ for homogeneous $f$ such that $D_+(f) \cong \text{Spec} \ S_{(f)}$.

Sheaves of modules $\tilde{M}$ on $\mathsf{Proj}$ are described in the same way as the structure sheaf $\mathcal{O}$: for a graded $S$-module $M$, we define $M_{\mathfrak{p}}$ in the analogous way and look at stalk-functions $m : U \to \coprod_{\mathfrak{p} \in U} M_{(\mathfrak{p})}$. A feature that distinguishes $\mathcal{O}$-modules from $\mathcal{O}_X$-modules on $X$ affine is existence of Serre twists. For $n \in \mathbb{Z}$, we can twist the grading of $S$ by making the $m$th component of $S$ into the $m+n$th component of $S(n)$. This twisting is compatible with localisation, so it extends to give twists of $\mathcal{O}$. Finally, we get twists of $\mathcal{O}$-modules by setting $\tilde{M}(n) = \tilde{M} \otimes \mathcal{O}(n)$. Structurally, $\tilde{M}$ and $\tilde{M}(n)$ are quite different sheaves: the former has local and global values determined by the degree 0 parts of $M$ and its localisations, while the latter has local and global values from the degree $n$ parts.