Split is the port from which the boats leave for the islands. There are day trips a plenty to be had. Try going to Hvar and renting a small boat with an outboard motor to visit the nearby islands. One of the closest ones had nude swimming (which utterly entranced 4 year old DD). Soltar, Vis and Brac are also worth a day trip.
Trogir is a local bus ride away. For the Dr Who fans, the episode with the vampires was filmed there.
The one sandy beach in Split is fantastic for families. It gets crowded, but there is practically no tide and it stays shallow a long way out. DD would happily get involved in long sand castle construction projects with children speaking a dozen languages.
Krka is a day trip on a bus. A beautiful forested area with river pools to swim in.
We booked a river kyaking day in the tourist information bureau in the centre of Split. We were picked up early in the morning, driven out to the river, provided with kyaks, life jackets etc, then at the end we were provided with a picnic and toys for the children (badminton stuff, some balls etc) It was a great day, we made friends with some English speaking Croats with children around DD's age and ended up walking to a local horse riding centre for a talk to the animals before being taken back home.
Take shoes to wear in the sea. Apart from the one sandy beach, the shore is all rocky. However, there's not much of a tide, the water is warm and there's not much in the way of waves.