linear algebra - process of finding eigenvalues and eigenvectors

I need to find eigenvalues/eigenvectors of different kinds of $n \times n$ matrices. For example, how would I determine these for the matrices listed below? What is the typical process? Should I always go by the route of finding eigenvalues by finding roots of characteristic polynomial and then getting eigenvectors by solving $(\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = 0$?

$\begin{bmatrix}
2&0&0\\ 1&2&0\\
0& 1 & 2
\end{bmatrix}
$

$\begin{bmatrix}
4 &1 &1 &1 \\

1&4 &1 &1 \\
1&1 &4 &1 \\
1& 1& 1& 4
\end{bmatrix}$

These are just examples. Typically I want to find eigenvectors of $n \times n$ matrices. If you can show me the process of finding solution of one of these matrices, that would be helpful.