One of the most interesting places I went to in Cartagena was el Palacio de la Inquisición (Palace of the Inquisition), which is now a museum. It houses a permanent exhibit on the Spanish Inquisition on the ground floor as well as other historical exhibitions and rotating displays on the upper level.
I take a big interest in the weird and the macabre, so obviously this place was right up my alley! They have a great collection of old torture devices with descriptions of each one listing how they were used and as punishments for what crimes. (FYI: The plaques with the descriptions are all in Spanish, and while you can pay extra for a guided English tour I chose not to.) What I could understand was pretty gruesome, and what I couldn’t was maybe better left that way!
The second floor of the Palace has several rooms with different exhibits on the history of Cartagena — the slave trade, pirate attacks — but most of it was large blocks of text (in Spanish, of course) so I’m not sure how much I got out of it.
When I was there (January 2013), they also had a temporary exhibition by an artist named Mauricio Vélez: mostly high-art nudes with heavy religious and sexual themes. Some of them involved extensive body painting and collaborations with other artists. Very striking images. That was actually probably my favorite part of the museum, and it was totally unexpected! So definitely check out what temporary exhibits they have going on when you visit.