Lackford Lakes is the place.
Although you cannot bring your dog, you can take a walk around and see nature as it should be with beautiful flowers and amazing wildlife. There are otters, ducks and dragonflies!
They have regular events and courses for adults and children. These cost money but other wise it is free to visit.
This is a great place to bring the children, they can watch the feeding stations from the visitor centre which also has a children's area to enhance their learning. There are also circular walks that are available to stretch the little legs and tire them out.
All the Best!
American to Britain