So I have had these questions confusing me for some time and tried googling but came up with long , hard-to-understand equations and notations and articles. So I would be glad if someone could clarify these things a bit more simpler.

Question

I know (mass) moment of inertia = integrate(r^2) dm , where r is the perpendicular distance from the axis of rotation to the mass center of the elementary mass.

Product of inertia is the product of perpendicular distances to the same in two axis.
eg : Ixy, Ixz

What do we need to do to calculate principal moments of inertia. Does that mean finding Ixx, Iyy, Izz?

what do we need to do to find the direction of the principal axes? Does that mean finding the inertia matrix I,
( Ixx 0 0
I = 0 Iyy 0
0 0 Izz)

or?

Answer 1

Yes.
If you have an moment of inertia matrix that has off diagonal numbers then rotation about any of those axes will cause wobble like an unbalanced car wheel. Think about trying to spin a dumb bell around an axis 30 degrees off the one through the two weights at the ends. The trick is to rotate the axes of spin until the three axes of rotation line up with the principal moments of inertia. Perhaps what you are looking for is this:
https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

Answer 2

Ah ha
Here is a more basic rationale and they even usee my dumb bell!

http://www.physics.arizona.edu/~varnes/Teaching/321Fall2004/Notes/Lecture34.pdf