Which eigenvalues and eigenvectors of a linear system are of interest and what to do with them? Given a linear system of equations, is there any "general" criteria how to select eigenvalues and eigenvectors after computing them? For example, positive or negative, the biggest or the smallest, complex or real? 
And I'm also asking myself what is the next step after computing the Eigenvalues/Eigenvectors, that is, how to analyze the system any further. So what would a mathematician do normally?
 A: As Henning Makholm observes in the comments, in practice when you compute eigenvalues and eigenvectors of a linear transformation, you already have some goal in mind. Often, in my experience, this goal is the search for a basis of a vector space which is tailored to the linear transformation. 
In an arbitrary basis, a linear transformation looks like a random $n\times n$ matrix. However, if you know the eigenvalues and eigenvectors, you can decompose the linear transformation into its action on each eigenspace. The action on a single eigenspace is much simpler -- it must be some combination of rotation, dilation, and skewing, each of which is much easier to understand than a regular matrix.  This is the idea behind the Jordan normal form of a linear transformation. 
In the case of a differential operator acting on an infinite-dimensional space of functions, an eigenbasis makes solving differential equations formally easier. For example, consider the heat equation $u_t - \Delta u = 0$ on a manifold $M$. If we have an eigenbasis $v_k$ associated to eigenvalues $\lambda_k$, then in this eigenbasis, any function $u(t) = \sum_k a_k(t)v_k$. Putting this decomposition back into the heat equation,
$$ 0 = \sum_k a_k'(t)v_k - \sum_k a_k(t)\lambda_k v_k = \sum_k (a_k'(t) - \lambda_ka_k(t))v_k$$
so we immediately see that every $a_k$ is a solution to the equation $a_k' = \lambda_ka_k$, i.e., $a_k = e^{\lambda_k t}$. So we immediately know to look for solutions of the form $u(t) = \sum_k e^{\lambda_k t}v_k.$ (There's a lot of analysis needed to justify these computations, but the point is that thinking in terms of eigenvalues and eigenvectors gives the intuition behind it.)
One also searches for eigenvalues and eigenvectors of a linear transformation in order to better understand the operator. In addition to geometric information about stretching and invariant spaces, you can easily compute from the eigenvalues many important quantities. For instance, the trace of the transformation is the sum of the eigenvalues; the determinant, the product. 
If the linear transformation is a differential operator acting on a vector space of functions from a manifold $M$, in addition to making solutions to differential equations intuitively easier, its spectrum can contain important geometric and topological information about the manifold. For example, Weyl's law indicates that the volume of a Riemannian manifold can be extracted from the eigenvalues of the Laplace operator on the manifold. The trace formulae (Selberg, Duistermaat-Guillemin, and many others) relate the eigenvalues of a differential operator to the spectrum of geodesic lengths on the manifold. The lowest nonzero eigenvalue of the Laplace operator controls (via results of Cheeger and Buser) the size of "thin necks" in the manifold in a manner that can be made precise. A theorem of Cheeger and Muller indicates that the zeta-regularized determinant (product of eigenvalues) of the Laplace operator is related to the manifold's Reidemeister torsion, a topological invariant.
(Forgive me for getting excited; this is my research area.)
