Let $X$ denote the blowup of $\mathbb P^2$, $E$ the exceptional divisor, and $H$ the pullback of the hyperplane class. How can I compute $H^0(X,mE+nH)$, $H^1(X,mE+nH)$, and $H^2(X,mE+nH)$ for $m,n \in \mathbb Z$? If I'm working analytically, how can I think of these geometrically (e.g. "an element of $H^1(X,2E)$ is equivalent to a $1$-form on $\mathbb P^2$ such that...")?
I assume you're only blowing up at one point, so here's at least a nice geometric description for $H^0$.
If $p$ is the point blown up to $E$, then $dH-E$ is the system of plane curves of degree $d$ passing through $p$, $dH-2E$ are those that have a double point at $p$, etc. This works with any number of blownup points, and a good exercise is using this interpretation to find all 27 lines on a cubic surface (which is the blowup at 6 points)