First Officer reported fatigue; unfamiliar airport departure; time pressure; CRM breakdown; altitude overshoot with FMC error; resulted in ATC action for terrain avoidance and TCAS RA.

Narrative

Departing TRM; long day after a series of heavy duty and early body clock getups. Short flight scheduled. We were an hour late. Received a clearance with a 3 minute void time. We were feeling pressure to get airborne. Due to this and our general fatigue; we failed as a crew to do a comprehensive briefing of our departure procedure. It was a first time operating out of TRM. All of these factors should have made us act with care; but I believe operating out of an uncontrolled airport; it's a more casual operation. We didn't brief the Departure procedure. The graphic showed a line between TRM VOR and PSP VOR; our first fix. It led me to simply think that was our routing after takeoff. The procedure actually requires a turn to the south and climb. ATC caught our error promptly and re cleared us to DEMEY intersection. Though the Captain loaded the fix; the FMC and autopilot directed a turn to a course toward terrain. ATC was alert and vectored us away from the terrain. I disconnected the autopilot to make the turn promptly. My head was swimming and I became task saturated trying to fly the plane and figure out the Navigation issues. My altitude control was poor which lead to a momentary TCAS alert for VFR traffic. Needless to say; it was ugly.