What the Upanishads say about the self
The Upanishads describe the self by peeling rather than by defining. They work inwards from the body through layers, then through the states you pass through every night, and stop at something that cannot be described because it is the one doing the describing. Nine concepts here, from six different Upanishads. Two of them are images rather than arguments, and they carry the teaching better than the arguments do.
Atman - The True Self
Atman is the word for the self that sees, hears and knows, rather than the body that appears to do the seeing. The Prasna Upanishad places it behind every faculty at once.
Prasna Upanishad 4.9
The Five Sheaths - Pancha Kosha
The self is described as covered by five layers: food, breath, mind, understanding and bliss. It works as a method, since each layer is something you can name as yours.
Katha Upanishad 1.3.10
The Three States of Consciousness
Waking, dreaming and dreamless sleep are treated as evidence rather than as topics. Something is continuous across three conditions that share almost nothing.
Mandukya Upanishad 7
Turiya - The Fourth State
The fourth is not another state but what the other three appear in. The Mandukya will only describe it by negatives: not inward, not outward, not knowing, not unknowing.
Mandukya Upanishad 7
The Two Birds
Two birds sit in one tree; one eats the fruit and the other simply watches. Sorrow ends, the verse says, when the eating bird notices the other one.
Mundaka Upanishad 3.1.1
The Chariot Metaphor
The body is a chariot, the senses are horses, the mind is the reins, the intellect drives and the self rides. The picture makes one point: the passenger should be choosing the road.
Katha Upanishad 1.3.3-4
The Self in the Heart
The self is placed in the cave of the heart, meaning a small space within rather than the organ. These texts keep the infinite in the most intimate location available.
Mundaka Upanishad 2.2.1
Neti Neti - Not This, Not This
Not this, not this. Every description offered is refused, because anything you can hold in front of you as an object is not the one holding it.
Brihadaranyaka Upanishad 2.3.6
Self-Knowledge as Liberation
Knowing the self is not a step towards freedom; it is the freedom itself. One who knows Brahman becomes Brahman.
Mundaka Upanishad 3.2.9
You can read all of this as metaphysics, and many readers do. It also works as a plain instruction: look through your experience for whatever is not an object, and notice how little survives the search. The Upanishads say that what survives is not small.