Cartagena has a weath of history with many castles, Roman ruins, modernist architecture and a large number of museums. It also has a large port, which is a lovely area to stroll, take a boat trip, or sit out and have a coffee.
The old town area is easily accessible on foot though it is quite a steam climb up to the Castle. There is a light railway called the FEVE which runs into Cartagena from Los Nietos on the Mar Menor and through La Union which is an old mining town.
The "must see" locations in Cartagena are the port area, the Roman Ampitheatre and Castle and the Old Town with its attractive architecture and shops. There is a great article in the Independent with lots of ideas which is worth reading.