Hi, sorry for the delay in replying - your post seems to have been missed!
There's a play-park in Lazise (just outside the city gate if you walk along Via Porta del Lione) and another in Bardolino (in the park between Via San Severo and Via Fosse) which are great for young children. In Bardolino there's also a large, tree-shaded, grassy area near Punta Cornicello. If you walk northwards from the harbour and keep to the lakeside path you can't miss it. In August you may be grateful for some shade! You can get a bus to both Lazise and Bardolino from Peschiera:
Or you can take a boat - small children seem to enjoy travelling by boat around the lake:
You can stroll along the lakeside path between Lazise and Bardolino in about an hour, and between Bardolino and Garda in about 45 minutes.
Riva is at the north of the lake and a long way from Bella Italia, but it has lots of green, open space for running around. As far as I know, it's the only beach on the lake with a roped-off area and a lifeguard on duty.
Most of the lake beaches are shingle so your children may need beach shoes. There's a stretch of man-made sand near Lazise.