Terra Natura (Murcia Zoo) in Murcia is a good trip out for the little ones as is Rio Safari up near Elche/Santa Pola. Whilst Terra Natura is closer, I'd say that Rio Safari is probably slightly more suited towards younger children. Zenia Boulevard as well as the airports often have flyers with discount codes for Rio Safari.
Also Zenia Boulevard (if you want to take in a bit of shopping) has quite a large kid's play area with soft play and mini carousels etc.
Terra Natura has a water park attached, but that only opens for high season from June to start of Sept. There is also Aquapolis at Torrevieja, but again that is only open high season.
Of course, don't forget the lovely beaches and water of the Mar Menor - hours of beach fun for the children (who seem to be oblivious to slightly chilly waters - although it should be warming up nicely by then).
If you drive up to Lo Pagan you have the benefit of both the Mar Menor and the Med and a good walk out on the 'promenade' which is beautiful on a sunny late Spring day - you can have a go at playing with covering yourselves with the mud there or drive out to the wilder beaches and go flamingo spotting!
Cartegena has the tourist boat which takes trips out and around the harbour and it's good fun to see if you can get little ones to find a 'Roman Soldier' in Cartagena.
Hope some of that helps - enjoy your holiday