How do some traumatic experiences establish deep beliefs and ideas? Disclaimer - My terminology might be wrong.

How do some memorable experiences that are traumatic or tough, make the brain form some values. Is there a way to establish core values so strongly ourselves through any manual methods? (without needing to go through traumatic experiences)
Childhood experiences have a greater impact than events later on. It is cognitive structure forming time, with a still strong biological background. Also, the child might see itself as vulnerable, thus willing to form bonds with any protective thing that might appear (Giller, 1999):

Chronic early trauma — starting when the individual’s personality is forming — shapes a child’s (and later adult’s) perceptions and beliefs about everything.
Severe trauma can have a major impact on the course of life. Childhood trauma can cause the disruption of basic developmental tasks. The developmental tasks being learned at the time the trauma happens can help determine what the impact will be.

Children can recover to leading normal lives after traumatic events, but not all of them. It gets worst if there are continuous traumatic events (Giller, 1999):

Although most return to baseline functioning, a substantial minority of children develop severe acute or ongoing psychological symptoms (including PTSD symptoms) that bother them, interfere with their daily functioning, and warrant clinical attention. Some of these reactions can be quite severe and chronic.

We go with defense mechanisms to protect ourselves. They might get repressed in our memories but never really fade-out. It is not consensual that it is OK to break down these defenses and make people revive those traumatic experiences. Thus, i do believe you are in a psychodynamic therapy or others. And no, it is not the only way to &quot;reprogram&quot; the brain. There is also cognitive , behavioral, and cognitive-behavioral therapy, which are problem focused, making you relearn the correct way to think/behave in problematic contexts.

CBT for trauma includes: learning how to cope with anxiety and negative thoughts, managing anger, preparing for stress reactions, handling future trauma symptoms, addressing urges to 'self-soothe' with alcohol or drugs and communicating and relating effectively with people (National Centre for PTSD, 2008).
The CBT model when used with survivors of child abuse usually focuses on the 'here and now' rather than revisiting the trauma itself (Henderson, 2006).
Cognitive–behavioral therapy (CBT) techniques have been shown to be effective in treating children and adolescents who have persistent trauma reactions. CBT has been demonstrated to reduce serious trauma reactions, such as PTSD, other anxiety and depressive symptoms, and behavioral problems. Most evidence-based, trauma-focused treatments include opportunities for the child to review the trauma in a safe, secure environment under the guidance of a specially trained mental health professional. CBT and other trauma-focused techniques can help children with cognitive distortions related to the trauma, such as self-blame, develop more adaptive understanding and perceptions of the trauma.

I'd hint that in a context which triggers some childhood trauma (even if it is in a deep layered reasoning), it still keeps that memory being fed, forbidding its physical memory-disposal. Learning the correct reasoning will make these memories fade away eventually.