To finde the center of $D_4$ is there a nice/smart way to find the center of $D_4$? rather then going through every element?  

By its represantation, $$D_4=<a,b> | bab^{-1}=a^{-1} , a^4=e=b^2$$
Thus, $ba^2b^{-1}=a^{-2}=a^2$ so $a^2\in Z(D_4)$
From that we see that $Z(G)$ has at least $2$ elements but can not be $4$ as $G/Z(G)$ become cylic and can not be $8$ as it is nonabelian.
$$Z(D_4)=\{e,a^2\}$$
