For a sweeping panoramic view, stride up to Castillo de la Concepción through the peacocked gardens, or hop on the lift. The first part of the visit is all about the city vistas, before you pass through a worthwhile display on Cartagena's tumultuous history; there are several interesting audiovisuals with English subtitles.
From the pedestrian area, the castle is most easily reached via Calle Doctor Tapia Martínez.