You can hire bikes (family carts as shown above, or individual bikes) from various points within the park including; Viale Medici (Pincio), Viale J:W: Goethe, Viale delle Belle Arti and others.
You can rent a rowing boat at Giardin del Lago. Open daily from 9:30 am until sunset.
Take a trip around the park on the Cinecaffè Express train (shown below).
Take the kids to visit Casina di Raffaello (Viale della Casina di Raffaello- Piazza Siena), a recreation centre for children between the ages of 3 and 10. It offers a vast program of constantly revised activities. Some, aimed at creative play, are free; others have a cost. Open Tuesday- Sunday from 9am- 7pm. See http://www.casinadiraffaello.it for more details.
You could take the kids to the Cinema dei Piccoli, the smallest operating movie threatre in the world. With 63 seats, the climate controlled room offers a rich selection of films for children during the day (for adults in the evening). It is also wheelchair accessible. For more information visit http://www.cinemadipiccoli.it
There are also many sites, (like the one shown in the above photo) throughout the park with rides and activities for children. You can also rent roller or inline skates, pony rides and much more...