Fractional captain reported receiving a low altitude alert from ATC during approach descent.

Narrative

While flying the RNAV Y XX aircraft did not intercept the inbound course. Flying pilot disconnected the auto pilot and turned to re-intercept the inbound. ATC inquired and the Pilot flying elected to go around. We then came around and attempted the Localizer XX approach. The Flying pilot didn't initiate a descent to the next lower altitude. He then disconnected the auto pilot to descend. We overshot the next lower altitude and set off a terrain warning to ATC. They inquired and I replied that we were now VMC and had the runway environment in sight. We landed with no further incidence.Suggestions: It was a very short leg and I was very compressed. We initially had the Localizer XX approach selected. ATC said we could shoot either so we decided to input the RNAV Y XX. I suspect I may have loaded the one on top of the other. Then when we didn't intercept; we again loaded the Localizer again. This time mine appeared to load correctly but the fling pilots didn't. We corrected his and continued into VMC conditions and landed. I think as I gain more proficiency in this aircraft; I will be able to better troubleshoot procedural errors as such.