I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or $\mathbb{CP}^2$ to complement the discussion there. So I tried to work things out for the latter of these myself. Maybe someone could point me at a decent source for examples or tell me whether my very pedestrian calculations make any sense. Thanks in advance.
1) Some notation. Consider $\mathbb{CP}^2 \subset \mathbb{C}^3$. For $z_i \in \mathbb{C}^3$ a point in $\mathbb{CP}^2$ is given by $(z_1: z_2: z_3) \sim (\lambda z_1:\lambda z_2:\lambda_3)$. An open covering is given by $U_i = \{(z_1:z_2:z_3) \vert z_i \neq 0\}$. Furthermore there are maps $\phi_1 : U_1 \to \mathbb{C}^2$, $(z_1:z_2:z_3) \mapsto (z_2/z_1, z_3/z_1) \equiv (w^{(1)}_1, w^{(1)}_2) \in \mathbb{C}^2$. We define maps $\phi_{2,3}$ and coordinates $w^{(2,3)}_j$ identically. I.e. the $w^{(i)}_j$ are coordinates in a patch $U_i$. On $U_i \cap U_j$ we can consider $\phi_{ij} \equiv \phi_i \circ \phi_j^{-1}$.
2) With this out of the way, consider the divisor $D_1$ defined (globally) by $\{ z_1 = 0\}$. In other words, on $U_2$, $D_1$ is defined by $w^{(2)}_1 = 0$, on $U_3$ by $w^{(3)}_1 = 0$. Now, we want to link this to a (global) section of $\mathcal{M}^* / \mathcal{O}^*$ where $\mathcal{M}^*$ is the sheaf of not identically zero meromorphic functions and $\mathcal{O}^*$ is the sheaf of non-zero holomorphic functions. On $U_{2}$ and $U_{3}$, $D_1$ is locally defined by the zeros of the functions $g_{2} \equiv w^{(2)}_1, g_{3} \equiv w^{(3)}_1$. Since $D_1 \cap U_1 = \emptyset$, we can pick $g_1 = 1$. Direct calculation shows that on the overlaps $U_i \cap U_j$ the quotients $\frac{g_i}{g_j} \in \mathcal{O}^*(U_i \cap U_j)$ and thus we have constructed our global section.
3) The next step is to construct the associated line bundle $[ D_1 ]$. Since $g_{ij} \equiv \frac{g_i}{g_j} \in \mathcal{O}^*(U_i \cap U_j)$, the $g_{ij}$ define transition functions / cocycles for a line bundle on $\mathbb{CP}^2$.
4) Finally we construct a global section of $\mathcal{O}(D_1)$. To do so we consider a generic polynomial in the patch $U_1$, $\sigma_1 = \sum_{i,j \geq 0} a_{ij} (w_1^{(1)})^i (w_2^{(1)})^j$ (where $\sigma_1 = \sigma \vert_{U_1}$ for a hypothetical global section $\sigma$). Using both the transition functions $g_{ij}$ as well as the earlier $\phi_{ij} = \phi_i \circ \phi_j^{-1}$ one finds that $$ \sigma_2 = w_1^{(2)} \sum_{i,j \geq 0} a_{ij} \left( \frac{1}{w^{(2)}_1} \right)^i \left( \frac{w^{(2)}_2}{w^{(2)}_1} \right)^j. $$ A similar result for $\sigma_3$ restricts the $a_{ij}$ such that a global holomorphic section is defined by $$ \sigma_1 = a_{00} + a_{10} w^{(1)}_1 + a_{01} w^{(1)}_2 $$ and similar in the other patches.
5) There is a simple generalisation to $n D_1$ as well as $D_{2,3}$. In the former case we have to garnish a number of expressions with exponentials ${}^n$.
6) Question: How does the torus $T^2$ action on $\mathbb{CP}^2$ act on the various line bundles $n D_i$?