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?  

I think you can use the following fact:

Let $G$ be a finite group with center $Z(G)$, if the quotient $G/Z(G)$ is cyclic, then $G$ is abelian.

We obtain that the only possible cardinalities for $Z(D_4)$ are 2 or 1.
Now, you have only to show that there's a non trivial element such that commutes with all the other ones in $D_4$, but this is very easy
(Hint: if $\rho$ is such that $\rho^4=e$, look at $\rho^2$).
