Air carrier pilot reported receiving a terrain caution and pull up warning during final approach when the pilot flying went below the minimum safe altitude. Flight crew climbed and continued approach.

Narrative

At approximately XA:30 we started our decent into ZZZ planning for an ILS approach to runway XX on a light wind; clear; VFR night. The CA was Pilot Monitoring and the FO was Pilot Flying for the flight. We had briefed the ILS and I had emphasized the MSA for that quadrant was 3600 but failed to notify the FO that it was a mountainous airport and we can only accept an ILS approach. We were given a decent to 4000 on the downwind: abeam the airport ATC asked if we had the airport in sight to which we initially replied 'looking'. Abeam the Final Approach fix ATC cleared us for the visual. I was not anticipating the Pilot Flying to turn the airplane directly on a base leg directly to the Final Approach fix while starting a decent of about 1200 per minute. I glanced at my approach plate and noticed the minimum altitude crossing the IAF was 3000 and the step down was 2400 to the FAF which we were in between. I reiterated that the MSA was 3600 and saw we were decending throught 2800 then 2 seconds later we heard the Terrain Caution Aural warning. the Pilot Flying arrested the decent and started a shallow climb which silenced the alarm for a few seconds however; the Terrain Terrain Pull Up Warning sounded briefly so he immediately climbed while maintaining visual reference to the runway. once at a safe altitude we fully configured the airplane and continued to land.