What did the Upanishads teach?

He taught that it is in our nature to experience suffering, but that there is a way to experience life differently, by understanding the origins of our suffering.

These origins of suffering have to do with clinging to things as permanent (such as the notion of a permanent self, among others), when in fact there are no such permanent things.

The way to overcome this clinging is to cultivate certain habits in how we experience the world. These habits include understanding how morality & remorse affect our experience, how our ability to concentrate on our experience from moment to moment affects it, and how developing wisdom about the true nature of reality affects it. This true nature of reality is normally hidden from us by mistaken thoughts & feelings. It is what exists beneath or behind the world of thought-feelings.

The habits of experience needed to penetrate the delusions of thought-feelings are developed by pursuing behavior that does not lead to remorse, by increasing one's ability to concentrate & be aware of what's happening in the moment, & by thus developing the wisdom to live with true understanding of the nature of things are they actually are.