We go to Cartagena to visit family quite a bit, it’s about 20 minutes away, never taken my bike though, but while exploring the area there looks like loads of places to go riding, just get out & explore. I’ve just sent my father in law an email to ask him about routes etc, as a few of his neighbours go out for the day into the mountains.
Generally, what are you looking to do out there? The Mar Menor is a cracking beach for kids, if you have any, it’s man made & you can paddle out as far as you like & never get out of your depth, a few nice bars & restaurants there to. Around an hours drive to Benidorm, if you fancy that kind of thing.