You can easily walk from the train station once in Cartagena - there's lots of history which you can walk around without getting taxis etc as long as you're ok to walk.
As ToffeeCambs says,
by train: http://www.renfe.com/EN/viajeros/index.html (in english)
by bus: http://www.movelia.es/en (in english)
You can only view bus schedules up to early September at the moment as all the schedules will change to autum/winter schedules in September.
Try to get the quicker bus if you opt to go by bus.
Note the times of the 1st and last trains on weekends.
The Murcia city centre bus station is here: https://goo.gl/maps/RBmT57zmXGA2
Thank you guys, you have been most helpful, its appreciated. :)