Index Notation & Dirac Notation

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Why do I solve the Schrodinger Equation for? Why are Energy levels important? etc. I know nothing about the theory behind QM. I know only how to calculate questions that come up.

Schroedinger's equation is the notional counterpart to Newton's second law. It is the equation of motion. In classical mechanics, the state of a particle is specified by its position and momentum. If you know the state at one time, Newton's second law tells you the state any other time. In quantum mechanics, the state is a unit vector (more strictly speaking, the state is a ray that can be represented as a unit vector), such as the wave function. If you know the state at any time between two particular measurements, Schroedinger's equation tells you the state at any other time between those two particular measurements.

Energy levels are important for two reasons. The first reason is that energy levels can be measured by spectroscopy, so checking that quantum mechanics predicts the energy levels observed is an important check that quantum mechanics is correct.

A second, more technical reason that energy levels are important is that they determine the time evolution of a wave function that is an energy eigenstate. The time evolution of any wave function can be written as the superposition of the time evolution of individual energy eigenstates. But this technical reason is just applied mathematics, not physics.