A visit to the Alcazar, a Muslim palace that was taken over and expanded by Christian kings, is a highlight of any trip to Sevilla. Even our toddler said it was his favorite outing in Seville. The lush interiors are fascinating and beautiful, and children can run freely in the beautiful gardens. The flow of the palace itself alternates between indoors and outdoors, making it even easier for active kids to enjoy. If your kids are particularly in awe of princes and princesses, they might enjoy a tour of the (lived in) royal apartments upstairs. Ask for details at the ticket desk when you enter the Alcazar.
There is a pond with ducks and fish, so bring along some crackers or bread (these animals are by no means wild)
Avoid the lines by visiting later in the day. If you have very young children, ask a security guard whether it is ok to skip the lines.
There is a playground outside the walls.