How can I spot if two matrices are similar by just looking at them? Is there any simple way to do this?  The normal process of finding a diagonal matrix takes too long. 
 A: So let's take two arbitrary matrices $\begin{pmatrix} a& b \\c& d\end{pmatrix}$ and $\begin{pmatrix} e& f \\g& h\end{pmatrix}$. You should know the following result:
If two matrices are similar, then they have the same eigenvalues.
The converse is not true, however the necessary condition is quite strong itself in the $2*2$ case. There are only two eigenvalues, they are determined by the trace of the matrix (sum of diagonal entries, and also of eigenvalues) and determinant of the matrix (product of eigenvalues).
In this case, that means at least that:
$a+d=e+h$, $ad-bc=eh-gf$. If one of these two is not satisfied, your matrices are not similar.
If the condition above is satisfied, it's still possible that the matrices are not similar. for example, $\begin{pmatrix} 1& 0 \\0& 1\end{pmatrix}$ and $\begin{pmatrix} 1& 2 \\0& 1\end{pmatrix}$ have the same eigenvalues, but are clearly not similar, because the identity is only similar to itself.
Your inspection stops here, unfortunately. This page: How do I tell if matrices are similar? will be of help to you from here, where the techniques used do not involve inspection, unfortunately.
