Looking at both Matsuki's "Introduction to the Mori Program" and Reid's "Chapters on Algebraic Surfaces", it does not seem that there is a uniform and quick proof to abundance conjecture for surfaces. In both books, the Authors separately consider the three cases:
$\textrm{kod}(X)=2$. Then abundance is proven by explicitly constructing the canonical model $X^{can}$ of $X$, by means of contractions of the $(-2)$-cycles. Another approach is using Kawamata-Viehweg vanishing, if one wants to prove base-point freeness in a more general setting. Also, one could apply Bombieri's result, which ensures that $|5K_X|$ is always a birational morphism onto the canonical model for any surface of general type.
$\textrm{kod}(X)=1$. Then one must prove the existence of an elliptic pencil on $X$ and the canonical bundle formula for elliptic fibrations. These are both rather subtle results, and I do not know any proof avoiding them.
$\textrm{kod}(X)=0$. Then the proof is obtained by looking at the Albanese map, and the analysis needed is essentially equivalent to the Enriques-Kodaira classification.