The Botanic Garden is a serene kind of place, with lots of different plant life and trees to wander past, big lawns to lie on, and some interesting old buildings, statues etc. There's a nice cafe in the middle, and a lake meandering through - kids can feed the ducks (and the odd turtle!)
It's right inside the Adelaide CBD, and near the Botanic Hotel and national wine centre, so a good spot for tourists to see.