We know that blowing up a point on a surface produces a $(1)$ curve. Is there any such standard techniques to produce $(2)$ curves in a smooth surface?

In case you want a curve of arbitrary genus with arbitrary negative selfintersection, you can do this: Let $C$ be a smooth projective genus $g$ curve on a smooth surface. Suppose $C^2=n$ and take $m$ (pairwise) different points on $C$ and blow them up. The strict transform of $C$ is isomorphic to $C$, so it has genus $g$, and has selfintersection $nm$. In other words, you can achieve arbitrary combinations of genus and negativeselfintersection. 


Here is something that might help answer your question: Blowingup rational double points on normal singular surfaces produces $(2)$curves of genus zero. Conversely Artin [1, Thm. 2.7] showed that (under suitable conditions) every such $(2)$curve of genus zero arises in this way. [1] Artin  Some numerical criterion for contractability of curves on surfaces. 

