how much information do I need to know about a matrix to have an idea about what the matrix is doing?
When I took Linear Algebra, it was really helpful to visualize the geometry of course! This is a really worthwhile endeavor in any branch of mathematics. When you're given a matrix, keep in mind always that you're looking at linear transformation from one vector space to another. The matrix can be thought of as column (row) vectors, depending on your point of view. Rather than try (in vain) to describe all of the geometry contained within a matrix, I will point you in the direction to begin:
If you haven't yet, please look at MIT's open courseware Linear Algebra video lectures by Dr. Gilbert Strang. His lectures are very good. I haven't studied his book, but I've studied from other books (both cheap and very good: Shilov's Linear Algebra and Axler's Linear Algebra Done Right; buy/rent/obtain them somehow).
The point of eigen(german for characteristic)values is that these vectors remain the same(up to a scaling factor, $\lambda$, under the linear transformation. Take a linear transformation that rotates by 180 degrees in $\mathbb{R}^2$. The vector $(1,0)$ is mapped to $(-1,0)$. This is peculiar: it seems we have only multiplied this vector by a $\lambda = -1$. You can probably guess this is going to be an eigenvalue (actually two). $(0,1)$ is similarly mapped to $(0,-1)$. And in general a vector $(a,b)$ is mapped to $(-a,-b)$. This computation is much easier when the matrix is in its diagonalized form (two -1's on the diagonals).
The second question is if anyone knows fast method to compute eigenvalues/eigenvector in your head. I know people who can do this for any 2-be-2 matrix, but I don't know which methods are they using.
You can try $\det{(A-\lambda I)}=0$. This determinant is not hard, and how the polynomial splits should be recognizable. There are other methods, of course. Maybe you could provide an example matrix? (see Axler for Linear Algebra with little mention of the determinant...) Hopefully you remember that if you have a diagonal matrix $A$, the eigenvalues are on the diagonal :-).
I hope this helped a bit.
Edit: Ohh! I forgot! What Yuval said is very good for quickly finding the product and sum of the eigenvalues.