I am trying to understand wreath products but cannot find an example of one being computed in depth.
I know that $D_{4}$ is isomorphic to $\mathbb{Z}_{2} \wr \mathbb{Z}_{2}$, but I cannot seem to understand why this would be true from definitions I read in textbooks.
This wreath product is just the semidirect product of the Klein-4 group and $\mathbb{Z}_{2}$. How exactly would this semidirect product be computed, and how can I see that this is isomorphic to $D_{4}$?
The wreath product is defined in terms of a semi-direct product. There is the restricted and the unrestricted wreath product, depending as one takes a direct sum or product in the definition. These agree for finite products.
You need a group action. In this case we get $\Bbb Z_2$ acting on $\Bbb Z_2^2$ by shifts. Then it is the semi-direct product $\Bbb Z_2^2\rtimes\Bbb Z_2$. It turns out that this is indeed the usual $D_4\cong\Bbb Z_4\rtimes\Bbb Z_2$. (Notice that the actions in these two semi-direct products are different.)
$\Bbb Z_2\wr\Bbb Z_2$ is the smallest non-trivial wreath product. It also equals the hyperoctahedral group $S_2\wr S_2$.
To see the isomorphism, you could work through the notation; or you can convince yourself that this is not the quaternions, the only other nonabelian group of order $8$.