An air taxi Captain reported becoming unstable during a Visual Approach and received a Terrain Warning as a result.

Narrative

Flight into ZZZ. Using the Guided Visual Approach XX. It was busy environment with circumnavigating cumulus clouds and numerous aircraft on ATC frequencies. Initially we were given 14;000ft; prior to reaching ZZZZZ; and cleared for the Visual Approach. Because of the busy workload; we did not accomplish the approach checklist until being cleared for the visual. This was a distraction. We were slightly to the right of the inbound course. In correcting to get established on course; the VNAV did not capture; so we were high; and getting slow. As a precautionary measure; I took controls; and disengaged the autopilot to descend to get back on the vertical guide path; which we did prior to reaching ZZZZZ1. Did a slight S turn to make the recapture of the vertical guide path more smooth. Also; we had a Caution Terrain by ZZZZZ2; we were on path before that; and visual clear of the mountain top; and well stabilized. One additional note; flew for the last two days with my partner; in the mountain region of the West; with numerous thunderstorms; and my partner has been professional with very good skills as well as knowledge of flying with inclement weather. The chain of event started well before the approach began. The best course is to be better prepared for an approach; further out in mountain airports; accomplishing a checklist at 18;000; even on a good day; leaves very little time before starting the approach. It's hard to come up with a solution when distracted by weather.