Pskov-Caves Monastery, also called Pskovo-Pechersky Monastery, is one of the strongest day trips from Pskov. Its walls and towers make it look like a border fortress, but the heart of the place is religious: cave churches, old burial passages, and a monastic community with unusually deep continuity.
The monastery is in Pechory near the Estonian border, so it needs a proper excursion rather than a quick detour from the kremlin. Dress and behave as you would at an active Orthodox monastery, and check current access rules before setting out.