A proper answer to your question would require at least a semester of Physical Chemistry. I would like to touch on a couple of issues (very loosely speaking):
- We know from quantum mechanics that electrons have certain properties, which are couched in something called a wave function. Mathematically, wave functions may be complex (i.e. have real and imaginary components) It is impossible to "see" complex quantities. So there must be something else.
- In fact, the only thing that we can "see" is the electron density, and the electron density of an atom is spherical. The electron density is the square (actually complex modulus) of the wave function, and so a possible set of (perhaps imaginary) wave functions are the "Spherical Harmonics" (s, p, d, ... fuctions)
The only system that has an analytic solution is the Schroedinger equation for a hydrogen-like atom.
So what happens if we "cram" two or more atoms together to make a molecule? Is there an analytic solution for molecules that gives sigma, pi, etc.? Nope. But we can try to make an approximation, called the linear combination of Atomic Orbitals (s,p,d...) to give Molecular Orbitals (LCAO-MO)
The sigma and pi bonds (and anti bonds) are a "basis" or a representation of molecules that we construct from things we know everything about, the atomic orbitals of a hydrogen-like atom.
As a coarse analogy, when we watch TV, the pixels on the screen are a "basis set" which, individually, can be turned red green blue. Their combination gives rise to the image of whatever you think you see on the screen (State of the Union address, football game, etc.)
Simple fact is: two atoms in separation look like spheres. If they are pushed together, then their electron density forms a dumbbell shape. You can not look at the electron density and pick out sigma and pi features. There is no such thing as "hybridization" to form sp, sp2, sp3... These are convenient "lies" (linear combinations of primitive atomic orbitals) which sometimes get in the way of something so very simple: Electrons of Atom A are attracted to Protons in Electron B and vice versa. (This is a slight bend of the truth, because electrons are shared between both. But it is possible to identify atomic basins, please see Quantum Theory of Atoms in Molecules)
But to follow the conventional explanation: electrons "go" into bonding orbitals (e.g. sigma bonds) first because sigma bonds usually look more like s atomic orbitals. s atomic orbitals are primarily centered upon the nucleus, and since electrons and protons have an attractive interaction, that is the lowest energy configuration.