Chemistry - How does localized bonding theory and hybridisation work?

First of all, 'hybridisation' is a hypothetical concept, i.e. orbitals don't really mix to form new orbitals, but 'hybridisations' are very successful in explaining structures of molecules and their physical properties. So knowing that a compound has $\text{sp}^2$ hybridisation will just help you arrange the constituent atoms around the central atom. Physically it has no other meaning. It's just like vector cross product in Physics (for example the torque on a rotating wheel acts along the axis, i mean what does that even mean physically).

Secondly regarding how to know if an atom has a $\pi$ or a $\sigma$ bond: observing many structures(Lewis structures) and recognizing certain trends is the way out.

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Some rules of the thumb that will help you draw structures are as follows:
1. Identify the central atom as the one that can make the most bonds.
2. Hydrogen makes only one bond. If Oxygen is present in the same compound, the Hydrogen is generally attached to the Oxygen forming a $\ce{O-H}$ group.
3. Second period elements cannot expand their octets(i.e have more than 8 electrons in valence shell)
4. Mostly second period elements can make $\pi$ bonds.(That really solves part of your query). Although in some rare cases third period elements may also make $\pi$ bonds.
5. Never neglect the lone pairs(or non-bonding electrons) of the central atom.

Remember that these are only thumb rules and may be violated sometimes.