For beginners, I always recommend a technique called vipassana (Pali, “insight”), which comes from the oldest tradition of Buddhism, the Theravada. The advantage of vipassana is that it can be taught in an entirely secular way. Experts in this practice generally acquire their training in a Buddhist context, of course—and most retreat centers in the U.S. and Europe still teach its associated Buddhist philosophy. Nevertheless, this method of introspection can be brought within any secular or scientific context without embarrassment. The same cannot be said for most other forms of “spiritual” instruction.
The quality of mind cultivated in vipassana is generally referred to as “mindfulness” (the Pali word is sati), and there is a quickly growing literature on its psychological benefits. Mindfulness is simply a state of open, nonjudgmental, and nondiscursive attention to the contents of consciousness, whether pleasant or unpleasant. Cultivating this quality of mind has been shown to modulate pain, mitigate anxiety and depression, improve cognitive function, and even produce changes in gray matter density in regions of the brain related to learning and memory, emotional regulation, and self awareness.
The practice of mindfulness is extraordinarily simple to describe, but it is in no sense easy. True mastery probably requires special talent and a lifetime of practice.
Our relationship to our own thinking is strange to the point of paradox, in fact. When we see a person walking down the street talking to himself, we generally assume that he is mentally ill. But we all talk to ourselves continuously — we just have the good sense to keep our mouths shut.